An Israeli military spokesman, Lt. Col. Nadav Shoshani, posted on social media a photo of what he said was a house damaged in Katzrin and said, “There was no other target in the area other than a civilian neighborhood and kids on their summer vacation.”

He added: “Attacks against our civilians will not go unanswered.”

The tit-for-tat strikes, and the Israeli official’s threat of further retaliation, highlighted how months of diplomatic efforts have failed to ease hostilities along the Israel-Lebanon border. And they came as the Biden administration has intensified its push for a cease-fire between Israel and Hamas in Gaza in hopes of averting a broader Middle East war, although Israeli and Hamas officials have been cool to the latest U.S. proposal.

The sites of the most recent Israeli strikes, in a range of about 40 to 60 miles north of the Israel-Lebanon border in the Bekaa Valley, are deeper inside Lebanon than many of the near-daily attacks the two countries have exchanged since the war in Gaza began. Hezbollah, like other groups in the region backed by Iran, has been attacking Israel in solidarity with the Palestinians in Gaza, leading to the cross-border fire from both countries. Hamas is also backed by Iran.

The military said in a statement that it had detected secondary explosions after its strikes on Wednesday, which it said indicated that there were large amounts of weapons at the sites. At least three areas were targeted, including the town of Nabi Chit, according to Lebanon’s state-run news agency. There was no immediate comment from Lebanese officials on exactly what was hit.

On Monday, the Israeli military also said it had targeted a number of Hezbollah’s weapons storage facilities in the Bekaa Valley. Lebanon’s Health Ministry said those strikes injured nearly a dozen people, including two children.

Tensions have escalated sharply in the region in recent weeks since the killings of Fuad Shukr, a senior commander in Hezbollah, and Ismail Haniyeh, a top leader of Hamas. Israel has claimed responsibility for Mr. Shukr’s death and is widely believed to be responsible for Mr. Haniyeh’s. Hezbollah and Iran have vowed to retaliate more forcefully than before against Israel.

In a separate strike on Wednesday, the Israeli military said it had killed a commander in the Aqsa Martyrs Brigades, a militant group associated with the Palestinian Fatah faction that has fought alongside Hezbollah. The commander, Khalil al-Miqdah, who was killed in the strike in the southern Lebanese port city of Sidon, worked closely with Iran’s powerful Revolutionary Guards, the Israeli military said in a statement. That claim could not be independently verified, though the Aqsa Martyrs Brigades confirmed Mr. al-Miqdah’s death in a statement.

Secretary of State Antony J. Blinken toured the region this week to push for a cease-fire in Gaza, but there appeared to be no breakthroughs in talks. Officials familiar with the latest U.S.-backed proposal said it left major disagreements between Hamas and Israel unresolved.

On Tuesday, a senior Iranian military official, Gen. Ali Mohammad Naeini, the spokesman for the Islamic Revolutionary Guards Corps, suggested that an attack on Israel might have been placed on hold.

The Israeli police said that they had sent reinforcements to try to maintain order, and Israeli news media reported that officers had sealed off several streets, used water cannons to disperse crowds and beaten some protesters with batons. When asked about the response, the police said in a statement that officers had been “forced to act using various means” as protests continued and demonstrators broke through a blockade, with some protesters throwing water bottles. Five people were arrested, the police statement added.

The protest highlights the increased friction between Israel’s mainstream secular society and the ultra-Orthodox, the fastest-growing part of the population.

Some ultra-Orthodox Israelis do not fully recognize the state of Israel, rejecting secular Jewish sovereignty and military service. Many ultra-Orthodox see full-time Torah study as crucial, arguing that this scholarship is what has ensured the survival of Jews for centuries.

A debate long seen by the rest of Israeli society as one over equality has increasingly become one about security, too. Israel has been in a 10-month war with Hamas in Gaza, and skirmishes with the Lebanese militant group Hezbollah have intensified. Fear of a regional war still looms amid concerns that both Hezbollah and its patron, Iran, could launch retaliatory attacks for recent assassinations attributed to Israel.

A video from the Israeli broadcaster Channel 7 showed of one of the altercations at the protests on Wednesday, with one ultra-Orthodox demonstrator asking a counterprotester, “You want me to work for you?”

“Your protection is worthless,” the counterprotester says, in an apparent reference to the Torah. He soon punches the ultra-Orthodox man.

Israel’s military hoped to defuse tensions over the conscriptions of the ultra-Orthodox — some 4,800 are to be drafted this year — by focusing on unmarried male members of the community who are in the work force and not on religious students.

But last week, hundreds of ultra-Orthodox protested outside the Alon military base. Out of 90 ultra-Orthodox candidates for service who were summoned to the base that day, only 12 showed up and completed the processes required for conscription.

Tensions over the draft pose yet another test for the government of Prime Minister Benjamin Netanyahu, who has struggled to balance the demands of the ultra-Orthodox parties that form a critical part of his coalition with his own nationalist base, some of whom no longer believe the exemptions are viable.

Negotiations mediated by the United States, Egypt and Qatar have inched along since early December, when hostilities resumed after a one-week cease-fire during which Hamas released more than 100 people from captivity in Gaza and Israel freed 240 Palestinian prisoners. In late May, President Biden endorsed a new three-phase plan and the U.N. Security Council followed with a resolution supporting it.

The first phase would see a six-week cease-fire and the release of hostages in exchange for Palestinians being held in Israeli prisons. People displaced from northern Gaza would be able to return to their homes, many of which lie in ruins. During that time, Israeli forces would withdraw from populated areas of Gaza.

The second phase envisions a permanent cease-fire, while the third consists of a multiyear reconstruction plan for Gaza and the return of the remains of deceased hostages.

But for months, Israel and Hamas, whose negotiators do not speak directly to each other, have remained far apart on key issues.

What is the new U.S. proposal?

On Aug. 8, with the war in its 11th month, President Biden and the leaders of Egypt and Qatar said they were willing to present a “final” cease-fire proposal. Last week, at talks in Qatar, the United States presented what it called a “bridging proposal” to try to close some of the gaps between Israel and Hamas.

The details of that proposal have not been disclosed publicly, but the Biden administration has tried to put diplomatic heft behind it. Visiting Israel this week, Secretary of State Antony J. Blinken warned that this was “maybe the last opportunity” to secure a cease-fire, and later said that Prime Minister Benjamin Netanyahu had told him during a meeting that he was committed to the U.S. proposal.

But both Israeli and Hamas officials familiar with the talks said the U.S. proposal leaves major disagreements mostly unresolved.

What is the main sticking point?

In broad terms, the U.S. proposal appears to conform to new demands added by Mr. Netanyahu in July that some Israeli troops continue to patrol part of an area of Gaza along the border with Egypt, according to Hamas and Israeli officials.

This has emerged as a crucial issue. Mr. Netanyahu considers an Israeli military presence in the area, which Israel calls the Philadelphi Corridor and Egypt calls Salah Al Din, vital to preventing Hamas from rearming after the war or rebuilding tunnels to Egypt.

Mr. Netanyahu told a group including families of hostages this week that Israel would not withdraw from the border strip “under any circumstances,” the families said in a statement. Mr. Netanyahu’s office confirmed he had said that Israel would not withdraw from Philadelphi.

Hamas rejects a continued Israeli presence in the area and is demanding a complete Israeli withdrawal from Gaza. Egypt says that keeping Israeli troops in the Philadelphi Corridor would raise national security concerns and would be unacceptable to the Egyptian public.

Mr. Blinken told reporters on Tuesday that Israel has already agreed to terms of withdrawal and reaffirmed that the United States would “not accept any long-term occupation of Gaza by Israel.”

What are the other disputes?

Other areas of dispute have emerged publicly. Since the start of the war, Israeli forces have established what it calls a security buffer inside Gaza along its eastern border with Israel, demolishing Palestinian homes in the process, and it intends to keep a presence there after the war. Israel also wants to retain the option to return to fighting after the first phase of a cease-fire.

Israeli forces have built a security road, which they call the Netzarim corridor, that cuts across Gaza from east to west. Israeli officials have said they want troops to keep patrolling that road, through which Palestinians must travel between the north and south of the enclave. That would violate Hamas’s insistence on a complete Israeli withdrawal.