In the north, near Gaza City, at least eight people were killed and several others injured by Israeli bombardment of an orphanage building owned by al-Amal Institute for Orphans where hundreds of displaced civilians were staying, the institute said in a statement. A majority of those sheltering in the building, which was heavily damaged by the attack, were women and children, the institute said.

And across the enclave, the Israeli military said it had bombed four school buildings during the day. The strikes killed at least 17 people at a school east of Gaza City, and at least five people at a school sheltering displaced people in Nuseirat, in central Gaza, the Palestinian Civil Defense said.

The Israeli military did not immediately respond to questions about the locations of the other two schools. The military said, without providing evidence, that all four schools were being used as Hamas command and control centers — a claim it has repeatedly made to justify increasingly frequent strikes on school buildings in Gaza.

The strikes on the schools and orphanage on Wednesday were condemned by the French Foreign Ministry, which said in a statement that the Israeli forces had “repeatedly targeted civilian infrastructure where people are seeking refuge.” Action on Armed Violence, an advocacy group that focuses on the effect that conflict has on civilians, said in a new analysis on Tuesday that, on average, explosive Israeli weapons had hit civilian infrastructure in Gaza every three hours since the war began.

“We just want a normal year,” said Sigalit Orr, a tech consultant who lives in Hod Hasharon, a densely populated Tel Aviv suburb where more than 100 homes were damaged by Iran’s missile attack on Tuesday.

Israel’s prime minister, Benjamin Netanyahu, issued a short, grim video message just before the holiday began, expressing his condolences to the families of the fallen Israeli soldiers, but also warning that the conflict was far from over. “We are in the middle of a tough war against Iran’s axis of evil, which seeks to destroy us,” he said. “This will not happen.”

On Tuesday night, Iran launched about 200 ballistic missiles at Israel in what it said was retaliation for assassinations of top leaders of its proxy groups, including those of Hamas’s political chief, Ismail Haniyeh, in Tehran in July, widely attributed to Israel, and of Hezbollah’s leader, Hassan Nasrallah, in an airstrike near Beirut on Friday.

Addressing the nation, Mr. Netanyahu vowed that Israel would return about 70 living hostages still being held in Gaza nearly a year after the Oct. 7 Hamas-led attack that sparked a devastating war in the enclave. He also promised to return the more than 60,000 residents of northern Israel displaced from their homes after Hezbollah, in solidarity with Hamas, began firing rockets into Israel on Oct. 8, sparking what has become another war.

“We will ensure the eternal existence of Israel,” Mr. Netanyahu concluded.

The comments offered little solace to some.

“The feeling here is very bad. We can’t celebrate any holiday,” said Miri Gad Messika, a member of Kibbutz Be’eri in southern Israel, a community near Gaza that was hit hard by Hamas’s attack last year.

“We can’t celebrate when our friends and family are being tortured in Hamas tunnels,” she said. About 100 hostages are still being held in Gaza, about a third of whom Israeli authorities have estimated are dead.

But Ms. Gad Messika, a mother of three whose family fled their home on the kibbutz on Oct. 7 by jumping out a second-story window as the house was being stormed, insisted that she still believed in humanity.

She said her youngest daughter, 10, had not been able to sleep in her bed alone since last October; her 15-year-old could not concentrate in school, though she had once been an excellent student; and her 16-year-old son still limped after breaking his leg in their escape. But she is hopeful that they and her community will recover.

“When trauma happens, there are two choices: Stay ill, or grow,” she said, adding, “I’m optimistic. I have to be. I want to wish everyone a better year.”

That sentiment was echoed by others in Israel, a small country where everyone has been touched by the year of escalating fighting.

Since Oct. 7, Israelis have been living in a “manic depressive state,” alternating between despair and normalcy, Ms. Orr said, with frequent reminders that the situation is grim. “You’re happy and then you hear about something horrible that happened to someone close to you,” she added.

Ms. Orr, who celebrated Rosh Hashana with her sister and children in a small, muted gathering, noted that this was not at all a typical holiday season for Israelis. Normally, she joked, half of Israelis would have traveled abroad at this time and the other half would be on the road inside the country, hiking, staying in hotels and celebrating.

This year, there are few flights abroad, and travel in Israel has become nearly impossible. “We’re just within our houses,” Ms. Orr said. “It’s not the usual country and mood.”

This assessment was reflected in a video statement from Lt. Gen. Herzi Halevi, the Israeli military’s chief of staff, on Wednesday. “On the eve of the holiday, the I.D.F. is operating and striking on all fronts,” he said, referring to the Israel Defense Forces.

“We are fighting and we know that this holiday is not complete without the hostages,” General Halevi added. He also said that he mourned fallen soldiers and “embraced” their families and those wounded “who are paying the costly price of the war.”

Jonathan Dekel-Chen, the father of an Israeli American hostage, Sagui Dekel-Chen, said he did not believe that a widening war with Israel fighting on all fronts would solve the country’s problems. He has been in the United States in recent weeks to push for the revival of cease-fire negotiations to end the fighting in Gaza, which has killed more than 41,000 Palestinians, and to secure the release of hostages. That cease-fire effort faltered this summer and appears to have been abandoned entirely in recent weeks as Israel’s conflicts with Iran and its allied groups in the Middle East have broadened.

Mr. Dekel-Chen did not travel back to Israel to join his family for the holidays, saying there was no reason to celebrate this year.

“We find ourselves at this new year amid an escalating regional war that probably wouldn’t have happened without Oct. 7,” he said, adding that “both Hamas and Israel have to be forced to get to yes, and to do that, we need the world’s help so that this year won’t be as disastrous.”

After Israel invaded Lebanon to confront Iran’s strongest ally, Hezbollah, and Iran’s second massive missile attack on Israel in less than six months, Israel seems ready to strike Iran directly, in a much more forceful and public way than it ever has, and Iran has warned of massive retaliation if it does.

“We are in a different story right now,” said Yoel Guzansky, a former senior security official who oversaw Iran strategy on Israel’s National Security Council. “We have a consensus in Israel — among the military, the defense experts, analysts and politicians — that Israel should respond in force to Iran’s attack.”

To many Israelis, there is now little to lose: Iran’s efforts to strike the urban sprawl around Tel Aviv crossed a threshold that Tehran has never previously breached, even during its earlier missile attack in April, which targeted air bases but not civilian areas.

Critics of Israel often see the country as the primary instigator of unrest in the Middle East. But most Israelis see themselves as the victims of constant attack from Iran’s proxies — particularly Hamas in Gaza, the Houthis in Yemen and Hezbollah in Lebanon — and feel that they have not done enough to defend themselves. As a result, there are growing calls in Israel to make Iran fully accountable for its allies’ attacks, even if it risks an explosive reaction.

“Many in Israel see this as an opportunity to do more to inflict pain on Iran,” said Mr. Guzansky, who is now a fellow at the Institute for National Security Studies, an Israeli research group. “To make it stop.”

Israel has yet to make a decision about exactly how to respond, six Israeli officials and a senior U.S. official said, and the extent of its reaction will be affected by the level of support — both practical and rhetorical — provided by the United States. U.S. forces helped Israel shoot down incoming missiles from both Iranian attacks.

The exact nature of its response may not become clear until after Rosh Hashana, the Jewish new year holiday, which runs until sundown on Friday, according to the officials, all of whom requested anonymity to discuss a sensitive matter.

In talks with the Israeli government, the White House was expected to point to the relatively light damage caused by the Iranian missile attack on Tuesday and urge Israeli restraint, the U.S. official said. These pleas were expected to have little impact, the official added.

But Israel’s counterattack is expected to be far more forceful than its response to Iran’s first round of ballistic missiles in April, when Israel conducted limited strikes on an Iranian air defense battery and did not officially acknowledge its involvement in that attack.

Israeli officials have told their American counterparts that they think the response in April was too little and too restrained, according to the senior U.S. official. Israeli leaders feel they were wrong to listen to the White House’s urging at the time to conduct a measured retaliatory strike, the official said.

This time, Israel might target oil production sites and military bases, the officials said. Damaging oil refineries could harm Iran’s already frail economy, as well as send global oil markets into turmoil a month before the U.S. elections.

Despite media speculation, Israel is not currently planning to strike Iran’s nuclear facilities, according to four Israeli officials, even though Israel sees Iran’s efforts to create a nuclear weapons program as an existential threat.

Targeting nuclear sites, many of which are deep underground, would be hard without U.S. support. President Biden said Wednesday that he would not support an attack by Israel on Iranian nuclear sites.

Still, Israel’s response “should be seen everywhere. It should be felt by Iran. It should hurt Iran,” Mr. Guzansky urged. “In order to do that, you cannot hit a radar station again.”

Israelis were deeply shaken by the Hamas-led attack on Israel of Oct. 7 and its aftermath, an assault for them on the very idea of Israel as a haven for Jews.

Now, many have an increased tolerance for short-term danger in order to achieve long-term security, according to Sima Shine, a former senior intelligence officer who helped guide Israel’s Iran strategy. More Israelis want the government to do “things that we didn’t do in the past, because we cannot be under ongoing attacks from all sides,” Ms. Shine said.

“This is part of the miscalculation of all our enemies around,” Ms. Shine said. “They don’t understand what Oct. 7 has done to the Israeli people, to their willingness to take much more risks.”

For Israelis, Iran also now seems more vulnerable than it has for years. After Israel killed much of Hezbollah’s leadership in recent weeks and destroyed large parts of the group’s missile stockpiles, Iran can no longer count on meaningful support from its proxy in Lebanon if Israel conducts a more forceful attack on Tehran.

“Iran is much weaker than before,” said Mr. Guzansky, the former official. “Israel is freer to do more.”

According to the senior U.S. official, a sizable number of missiles failed on launch or before reaching Israeli air space, further exposing a vulnerability in Iran’s vaunted ballistic missile arsenal that was first revealed in April’s attack, which had an even greater failure rate.

Israel’s second successful defense against advanced ballistic missiles, coupled with Hezbollah’s weakened condition, will likely embolden Israel to opt for a more aggressive response than in April, the senior U.S. official said.

Recent comments from Israeli leaders give a sense of their growing ambition and confidence. Benjamin Netanyahu, the Israeli prime minister, directly addressed the Iranian public in a statement earlier this week, hinting at his support for regime change in Tehran.

Mr. Netanyahu also struck a particularly defiant tone after the attack on Tuesday night. “This evening, Iran made a big mistake — and it will pay for it,” he said. “Israel has the momentum and the ‘axis of evil’ is in retreat. We will do whatever needs to be done to continue this trend.”

To some experts, however, Israelis risk underestimating Iran’s resilience and overestimating their own ability to cause meaningful damage.

“The approach that Israel has had over 75 years has always been about hitting your enemy harder than you’re being hit,” said Andreas Krieg, an expert on warfare at King’s College, London. “That doesn’t work with a regime like Iran, though. I don’t think you can deter them.”

As a result, Israel may be able to inflict short-term damage but fall short of long-term change, he said.

“I still haven’t seen a strategy,” Mr. Krieg said. “That’s what I’m trying to get from the Israelis: What is your strategy to weaken the regime?”

Not only can ballistic missiles fly farther and have a greater strike impact than rockets, they also can reach their targets in minutes, even when launched from thousands of miles away.

The increased use of ballistic missiles and the possible introduction of the Fattah-1 meant that Tuesday’s missile assault “appears to have been more significant,” than Iran’s last attack on Israel in April, said Fabian Hinz, a missiles and Middle East expert at the International Institute for Strategic Studies in London.

Iran launched around 200 ballistic missiles on Tuesday, according to Israeli officials. By contrast, in April, Iran fired more than 120 ballistic missiles, along with about 30 cruise missiles and 170 drones, at Israel, nearly all of which were intercepted by Israeli and American air defenses.

Many of the Iranian missiles on Tuesday were also intercepted, although some fell in central and southern Israel, and it was not yet clear if a higher percentage of the ballistic missiles made it through air defenses than in April.

In a briefing on Wednesday, the Israeli military did not provide that information.

But a senior U.S. official said Israel’s defense against such a large barrage of ballistic missiles was a success, given that the Israelis could shoot down so many themselves. That official said the Israelis let some of the missiles land in Israeli territory because they knew they were going to land in unpopulated areas.

The Pentagon’s spokesman, Maj. Gen. Patrick S. Ryder, said U.S. Navy destroyers in the region fired a dozen SM-3 interceptors that took down a “handful” of the missiles.

The Islamic Revolutionary Guard Corps in Iran said in an initial statement that Tuesday’s strikes targeted “important military and security targets.” It later said Iran had hit some Israeli air force and radar bases, and destroyed missile defense radar systems, although that claim could not be independently confirmed. The Israeli military said in a statement on Wednesday that an unidentified air force base had sustained “a few hits” but that “essential infrastructure” was spared.

The April attacks hit an Israeli air force base in the Negev desert, which officials at the time said suffered only light damage and remained functioning.

That Iran relied only on ballistic missiles in Tuesday’s attacks “could indicate a deliberate strategy to minimize the warning and response time available to Israeli defenses,” Mr. Hinz said.

Among the debris seen in footage, he said, were spent rocket boosters that could have been for the Fattah-1 missile, which is designed with several features intended to enhance its survivability against missile defense systems.

Mr. Hinz also said the spent rocket boosters, instead, could have supported a Kheibar Shekan missile, which Iran has used since 2022 and is also able to evade at least some traditional air defense systems.

Mr. Hinz said footage of the strikes show evidence of numerous maneuvering warheads that can change their trajectory in flight and that were not intercepted by Israeli defenses for reasons that are not yet clear.

Israel relied on its Arrow 2 and Arrow 3 air defense systems to intercept the Iranian strikes, according to Yehoshua Kalisky, a military technology expert at the Institute for National Security Studies in Tel Aviv. He said they are the only Israeli systems that can intercept long-range ballistic missiles.

While Israel is believed to have a healthy stockpile of interceptors, Mr. Kalisky said the Arrow systems were synchronized with American air defenses to ensure that Israel would not be overwhelmed by the Iranian barrage.

“Only a few penetrated, so they were not saturated,” Mr. Kalisky said on Wednesday.

One person died in the strikes, a Palestinian man in the Israeli-occupied West Bank who was hit by falling shrapnel from the missiles.

Experts have described Iran’s missile arsenal as among the largest and most diverse in the Middle East. Its missile production has surged over the past 15 years, as it has significantly improved the weapons’ precision, guidance and aerodynamics technology.

As of 2022, U.S. intelligence estimates believed Iran had stockpiled more than 3,000 ballistic missiles, some with a strike range of up to 1,240 miles.

According to the senior American official, some of the missiles Tuesday failed on launch or before reaching Israeli airspace, further exposing a vulnerability in the Iranian arsenal that was revealed in April’s attack.

Eric Schmitt contributed reporting.

But when a reporter asked whether he would support a decision to strike Iran’s nuclear facilities, Mr. Biden said he would consider that a step too far.

“The answer is no,” the president said as he boarded Air Force One on his way to tour damage from Hurricane Helene in South Carolina and North Carolina. “All seven of us agree that they have a right to respond, but they should respond in proportion.”

Military analysts believe it would be difficult for Israel to significantly damage Iran’s nuclear facilities, which are buried deep underground, without at least some support from the United States. That reality made the president’s comments on Wednesday a serious hurdle.

The message was a diplomatic echo of what Mr. Biden has been saying to Israel’s leaders for almost a year. During that time he has urged them not to let their justifiable anger at the actions of their enemies compel them to make military decisions that could have negative long-term consequences for the security of the Israeli population and the entire region.

But Mr. Biden has struggled to persuade Prime Minister Benjamin Netanyahu of Israel to exercise restraint in response to the surprise Hamas-led attacks on Oct. 7 that killed more than 1,200 people, as well as the provocations from Hezbollah and Iran.

The president and his aides have at times appeared overly optimistic about their ability to influence Mr. Netanyahu’s decision-making, only to be disappointed when the prime minister took actions that ran counter to their advice. Israel’s monthslong bombing campaign in Gaza, where more than 40,000 people have died, often came over the objections of Mr. Biden and others in his administration.

People familiar with discussions between Mr. Biden’s national security officials and their counterparts in Israel said on Wednesday that the Israeli government did not appear eager to dramatically escalate the conflict with Iran after Tuesday’s missile attacks.

The strike, which followed Israel’s killing of Hezbollah’s leader last week, was the second time in six months that Iran has directly attacked Israel. In April, Iran fired about 120 missiles and drones at Israel; Tuesday’s attack included about 200 missiles. In both cases, the United States helped Israel shoot down almost all of the incoming threats.

Some American officials have concluded that the Israelis seem less eager for a major confrontation than they did in April, when the Biden administration talked them down from a large military response. If that assessment is accurate, it could help prevent the Middle East conflict from growing.

But public statements from Mr. Netanyahu and private comments from others in his government suggest that the American officials might once again be overly optimistic, with Israel seemingly ready to strike Iran in a more forceful and public way than it ever has.

At the same time, Israel is continuing to battle Hezbollah forces in Lebanon.

Over the past several weeks, Biden administration officials have warned Mr. Netanyahu that the fighting there could devolve into a Middle East conflict that draws in not only Iran but the United States and other countries. But Mr. Netanyahu has pursued an aggressive military campaign in Lebanon that largely ignored that advice.

Mr. Netanyahu’s order to approve the assassination of Hassan Nasrallah, the Hezbollah leader, came even as the United States and a dozen other nations were calling for a cease-fire between Israel and Lebanon.

One senior U.S. official said the Biden administration’s understanding is that Israel’s military clash with Hezbollah in southern Lebanon is “local and targeted and temporary in nature.” The administration’s view is that Israel is not seeking to invade Lebanon or hold territory, the U.S. official said.

The Biden administration was still in contact with Israeli and Lebanese officials on Wednesday and was hoping to find a diplomatic solution to end the conflict and allow “people the ability to go back to their homes.” U.S. officials, however, were growing increasingly concerned that the broader war in the Middle East that Mr. Biden had warned about for months was becoming a reality.

Zolan Kanno-Youngs contributed reporting.

Tuesday’s vice-presidential debate was supposed to be the last set-piece moment of Campaign 2024 before the final sprint to Election Day. It was all but overshadowed by the transfixing images of Iranian ballistic missiles confronting Israeli defense systems in the darkened skies over Tel Aviv and Jerusalem.

Mr. Netanyahu vowed retaliation — “Iran made a big mistake, and it will pay for it,” he said — while Ms. Harris was steadfast in what she called her “cleareyed” condemnation of Iran, denouncing it as a “destabilizing, dangerous force in the Middle East.”

Her running mate, Gov. Tim Walz of Minnesota, was left to frame the issue of a looming regional war as a test of character.

“What’s fundamental here is that steady leadership is going to matter,” Mr. Walz said in the debate. “It’s clear, and the world saw it on that debate stage a few weeks ago. A nearly 80-year-old Donald Trump talking about crowd sizes is not what we need in this moment.”

But any dreams of a triumphant diplomatic breakthrough to end the hostilities, bring home the remaining Israeli hostages in Gaza or even get peace talks on track in the waning days of the Biden administration were atomized along with that hail of debris from shattered Iranian missiles raining over Israel.

It was a moment the Republicans were not going to waste.

“As much as Governor Walz just accused Donald Trump of being an agent of chaos, Donald Trump actually delivered stability in the world,” Senator JD Vance of Ohio, the Republican vice-presidential nominee, said on Tuesday night.

To be sure, American voters still put foreign policy well down the list of their highest priorities, far below the economy, abortion, inflation and character. But to Mr. Trump, the conflict in the Middle East is part of a much larger narrative that he has been weaving the entire campaign.

He has said repeatedly that feckless Democratic leadership in Washington has let events spin out of control on matters like Afghanistan, Israel and Ukraine abroad; the U.S.-Mexican border; and the price of groceries at home: All was peaceful and prosperous when he was in power, and his strong hands on the wheel would bring peace and prosperity back.

“Look at the World today,” Mr. Trump wrote on social media on Tuesday. “Look at the missiles flying right now in the Middle East, look at what’s happening with Russia/Ukraine, look at Inflation destroying the World. NONE OF THIS HAPPENED WHILE I WAS PRESIDENT!”

His narrative leaves out key events, including Iran’s shelling of U.S. military forces stationed in Iraq and the pandemic-driven economic collapse of his final year in office. But Mr. Trump’s politics have always been impressionistic, and the events of recent days are helping his cause.

Israel is nearing all-out war with its biggest regional adversary, Iran.

And the Biden administration now appears incapable of restraining Mr. Netanyahu, who got virtually everything he wanted from Mr. Trump when he was in the White House and is likely to relish his return to power.

Lara Friedman, president of the Foundation for Middle East Peace, said that Ms. Harris seemed to have just hunkered down, waiting for Election Day before making any affirmative effort to change the course of events in the Middle East and pressure Mr. Netanyahu to de-escalate.

It isn’t working — politically or diplomatically.

“She was in a bind from the beginning,” Ms. Friedman said. “If she gave an inch” toward criticizing the Israeli government, “she would be framed as anti-Israel or even antisemitic. Even if she doesn’t give an inch, she’s still being framed as anti-Israel or antisemitic.”

“So maybe it would be better to conceptualize and stand behind a defensible policy” in the region, Ms. Friedman added.

Ms. Harris’s bind is only growing worse. The anger that Palestinian Americans were feeling toward the administration — especially in the key swing state of Michigan — has now been joined by anger from Lebanese Americans, also concentrated in Michigan, who are decrying the indiscriminate bombing of their homeland.

Attila Somfalvi, an independent political analyst in Israel, said on Wednesday that Mr. Netanyahu actually had more political space for a diplomatic resolution of the tensions. He has expanded his government beyond the narrow, far-right coalition that put him back into power, and the killing of Hezbollah’s leader, Hassan Nasrallah, has increased his popularity.

“There’s a feeling of strength again; people are saying, ‘Look, the magician is back,’” Mr. Somfalvi said.

But for years, going back to the presidency of Barack Obama, Mr. Netanyahu has cultivated the belief among his supporters that the Democrats are the enemy, weak supporters of Israel at best, treacherous at worst. And now, with the Israeli right sensing a possible return of Mr. Trump, Mr. Netanyahu has no political incentive to help Ms. Harris.

“It’s pretty clear where the prime minister stands,” Mr. Somfalvi said. “All those fans of Netanyahu are very pro-Trump. It doesn’t matter what Biden and Harris have done over the last year. They say they need Trump.”

Earlier this year, that was not clear. Dennis B. Ross, a former presidential envoy to the Middle East, mused in an interview in March that Mr. Netanyahu might — just might — want the peace accord with Saudi Arabia that the Biden administration was trying to broker as his legacy, instead of the carnage of Hamas’s Oct. 7 attack on Israel. To attain that, he would have to accept a cease-fire in Gaza and a resumption of talks on autonomy for the Palestinians.

“A normalization deal with Saudi Arabia would serve both men,” he said of Mr. Biden and Mr. Netanyahu, while conceding, “the longer this goes on, given the political calendar, the less chance this can happen.”

That political calendar is now spent. Ms. Harris is likely to use the growing crisis in the Middle East to look resolute and presidential, as she did on Tuesday when she described joining the president and his national security team in the White House Situation Room to watch Iran’s missile attack on Israel unfold.

“Israel, with our assistance, was able to defeat this attack,” she declared. “Our joint defenses have been effective, and this operation and successful cooperation saved many innocent lives.”

With an uncertain season of Jewish High Holy Days beginning Wednesday night, with the first anniversary of the Hamas massacre of Israelis arriving on Monday, and with early voting already underway, Mr. Trump and Mr. Vance will take every opportunity to lay conflict at the feet of the Democrats.

“Donald Trump consistently made the world more secure,” Mr. Vance said in the debate. “When did Iran and Hamas and their proxies attack Israel? It was during the administration of Kamala Harris.”

Citing what it said were Western promises of renewed diplomacy to end the conflict, Iran had largely stayed restrained in the face of the July assassination in Tehran of Hamas’s political leader, Ismail Haniyeh and as Israel intensified the fight against Hezbollah over recent weeks, culminating with the killing of the group’s leader and a ground invasion in Lebanon.

But with no diplomatic solutions in sight, Iran found itself facing an inversion of its regional strategy: It had built the partnerships of its “Axis of Resistance” — with groups in Gaza, Iraq, Lebanon, Yemen and elsewhere — on the theory that they could fight on its behalf, avoiding a direct and more dangerous confrontation with Israel. Instead, it was Iran that had to strike.

Even facing the risk of blows from Israel that could threaten their power at home, Iran’s leaders have worried that the cost of inaction was higher: If Iran did not retaliate for Israel’s assassination last week of the Hezbollah leader Hassan Nasrallah, perhaps the most influential figure in its network, unease could spread among its partners.

“The strategic advantage at this point is that it maintains the loyalty of these groups across the region, and therefore maintains its influence,” said Maha Yahya, director of the Carnegie Middle East Center in Beirut, Lebanon’s capital.

That could prove useful in a war that drags on and offers more openings to Israel’s weakened enemies.

The Houthis in Yemen or the Shiite militias in Iraq have been relatively unsuccessful at striking major blows with missiles inside Israel, despite repeated efforts, but there are other levers they have not pulled. Iraqi militias, for example, could inflict damage on U.S. military bases inside Iraq. And Iran could gain support from U.S. rivals like China or Russia in a bid to make the regional chaos more costly for Washington, Israel’s most important ally, though it remains far from clear that they would do so.

Iran’s success in crafting an alliance across the region was ironically helped on by the United States’ invasion of Iraq in 2003, as its yearslong occupation spurred the formation of many Shiite Muslim militias, with support from the regional Shiite power, Iran. These militias helped Iran forge a chain of alliances across the region, linking it geographically with its most important and powerful ally, Hezbollah, in Lebanon.

Their potency seemed to culminate with Iran’s success, thanks largely to Hezbollah’s forces, in helping Syria crush a rebellion against President Bashar al-Assad. In more recent years, Hezbollah and Iran began fostering ties with the Houthis, offering it more leverage against its longtime regional Gulf rival, Saudi Arabia, with whom it has now established diplomatic relations. But now, under heavy pressure from Israeli attacks in the wake of the Hamas-led Oct. 7 assault, Iran fears that network is at risk of unraveling.

Though the situation has recently looked dire for Iran’s allies, in particular Hezbollah, many analysts predict the dynamic could change if Israel were to push forward with what it has called limited ground operations in Lebanon. Ms. Yahya called that possibility a “favor to Hezbollah,” whose roots in guerrilla warfare could make a troop presence costlier for Israel than attacks by air.

Reports from southern Lebanon underline such claims, with Israel announcing the deaths of eight of its soldiers by Wednesday.

“Israel cannot kill the octopus — they won’t be able to eradicate it,” said Kassem Kassir, a Lebanese expert on Hezbollah who is close to the group. “The fight will continue.”

Some experts see Iran’s latest attack more as a message than an attempt to inflict serious damage, though it did manage to strike an Israeli air base and near the headquarters of its intelligence services, the Mossad. At the same time, Iranian officials said they had informed Switzerland, a country they often rely on as an intermediary, when they launched their attack, allowing for some advance warning.

A similar attempt, however, failed at restoring deterrence in April, when Iran fired a barrage of missiles at Israel in retaliation for a strike on its embassy in Damascus — an attack it telegraphed well in advance, in what was seen as an attempt to avoid causing so much damage that escalation was inevitable.

Iranian leaders said they refrained for months from a second attack on the back of promises and gestures from the United States and Europe that Iran’s restraint would enable a cease-fire in Lebanon and Gaza, experts said, as well potentially restarting negotiations on a nuclear deal.

They now feel such overtures were “entirely false,” as Iran’s new president, Masoud Pezeshkian, said this week.

“The view in Iran is that the West is, at best, unwilling to restrain Israel, and at worst, has a direct hand in the escalation,” said Mohammed Shabani, an Iran analyst and editor at the independent news site Amwaj.media. “The latest Iranian operation against Israel, therefore, certainly is accompanied by a message that Biden should act to draw a line under things.”

And with officials in Israel, and Republicans in Congress, calling this a moment to go after Iran, experts say the leadership in Tehran may see few reasons not to rapidly pursue nuclear weapons.

Deterrence or no, Iran and its allies may feel there is now little left to lose, said Ms. Yahya.

“There have been increasing noises from different policymakers saying ‘regime change, new order in the Middle East’ — this is the language we heard in 2003 when the United States invaded Iraq,” she said. “And we can see the disaster that ended up in.”

“It’s unfathomable. You can’t grasp such a thing,” said Keren Doron, 54. The blast that shook her house was so strong, she said, that it cracked some of the walls and shook window blinds loose.

Like millions of Israelis, Ms. Doron, her husband and her daughter spent Tuesday evening huddled in fortified rooms, hoping and praying that the strikes would not harm them.

No injuries were reported, said Amir Kochavi, the mayor of Hod Hasharon, but more than 100 homes were damaged by the barrage on Tuesday night. He said that the authorities had yet to find a major metal fragment, but they found particles around the impact site containing explosives they believed to be part of a warhead.

Ms. Doron lives mere feet from the site, and on the footpaths of the homes that ring the park lay remnants of ravaged roofs and shattered windows. In all, Mr. Kochavi said, 15 homes in Hod Hasharon have been declared uninhabitable.

Maj. Zacai Shriki, a liaison from the Israeli military’s Home Front Command, said the city appeared to be the hardest hit by Iran’s attack.

Just a few feet from where the missile fragment landed, Ms. Doron’s daughter, Tom, 11, attends the elementary school that was damaged. One classroom’s ceiling had caved in, and windows across the small campus were blown out.

“It’s unbelievable that I stood right here yesterday, never thinking something like this would happen to my school,” Tom said.

One of the rockets severely damaged a school in the southern city of Gedera, but there were no reports of injuries.

Debris from one intercepted missile landed near Arad in southern Israel. On Wednesday, people climbed atop it.

Although most of the missiles were intercepted, falling debris shattered windows and glass in a building in Tel Aviv.

Sameh al-Asali was a Palestinian from the Gaza Strip who had come to live in the city of Jericho in the Israeli-occupied West Bank. On Tuesday, a fragment of an Iranian missile fell on him, making him the only person known to be killed in the attack targeting Israel.

Debris also fell in the West Bank city of Hebron, where Palestinians on Wednesday inspected the mangled metal of what appeared to be part of an Iranian missile.

The military did not specify where the soldiers were killed, saying only that they “fell in battle in southern Lebanon” on Wednesday.

Five of the eight were members of the elite Egoz Unit, which the military had said earlier in the day was engaged in “targeted operations in several areas of southern Lebanon,” including “close-range engagements” with Hezbollah militants.

“I would like to send my condolences from the bottom of my heart to the families of our heroes who fell today in Lebanon,” Prime Minister Benjamin Netanyahu said in a statement.

In a series of statements posted online, Hezbollah said it had fought Israeli soldiers on Wednesday in the border villages of Yaroun, Odaisseh and Maroun al-Ras.

The group also posted video that showed two military helicopters evacuating soldiers from a dry, grassy field, including two people on a stretcher. But it did not say where the video had been taken.

Maroun al-Ras, which was the scene of a major battle during Israel’s 2006 invasion of Lebanon, is roughly one mile from the Israeli town of Avivim, which Hezbollah said it had targeted with “a salvo of rockets” earlier in the day. Avivim was evacuated last year because of similar attacks.

In Yaroun, Hezbollah said it had detonated an explosive device on Wednesday afternoon, resulting in injuries to Israeli soldiers. The Israeli military did not comment on the report of injuries, and it could not be independently verified.

Al-Manar, a television network owned and operated by Hezbollah, said fighters from the group’s elite Radwan Force had ambushed Israeli soldiers near Odaisseh after they crossed the border from the Israeli village of Misgav Am.

Lebanon’s army — which is not a party to the fighting between Israel and Hezbollah — said in a statement that Israeli forces had crossed the border and traveled roughly a quarter of a mile inside Lebanon in the areas of Yaroun and Odaisseh, “then withdrew after a short period.”

The Israeli military identified the eight soldiers who have been killed as Capt. Eitan Itzhak Oster, 22, Capt. Harel Etinger, 23, Capt. Itai Ariel Giat, 23, Major Noam Barzilay, 22, Major Or Mantzur, 21, Major Nazar Itkin, 21, Sergeant Almken Terefe, 21, and Sergeant Ido Broyer, 21. It did not identify the wounded soldiers.