Since the start of the war in Gaza, Turkey has staked out a position that Turkish officials and analysts say is driven by support for the Palestinians, anger at the war’s high civilian toll and domestic politics. Turkey recognizes Israel diplomatically, unlike many other Muslim-majority states, and wants to play a role in ending the Gaza war, while its leaders simultaneously stand up for Hamas, a group dedicated to Israel’s destruction.

Sympathy for the Palestinians is widespread in Turkey’s otherwise polarized society. Many Turks are genuinely horrified by the vast destruction and civilian deaths, so politicians across the spectrum have more to gain from criticizing Israel than from speaking about its security concerns.

Officials in and close to Mr. Erdogan’s ruling Justice and Development Party also have personal relationships with Hamas leaders that go back many years. Mr. Erdogan knew Mr. Haniyeh personally. His foreign minister and former intelligence chief, Hakan Fidan, met Mr. Haniyeh often and sometimes passed messages to him from the United States, Jeffry L. Flake, the departing U.S. ambassador to Turkey, told reporters this week.

Last week, during a news conference with his counterpart from Montenegro, Mr. Fidan set aside his normally staid style to lash out at Israel.

“The perpetrators of the massacre in Gaza shouldn’t remain without punishment,” he said. “Those murderers should be held accountable sooner or later at international courts.”

He also criticized countries that send military aid to Israel. He didn’t name them, but the United States is by far the biggest supplier of weapons to Israel.

“It is pretty clear who is escalating the tension,” he said. “Stop the habit of sending the bill to the wrong place. The road to peace and calm in the Middle East comes through reining in the craziness of Israel.”

It was not immediately clear whether Mr. Abbas’s invitation to Ankara was part of a specific Turkish policy proposal. Mr. Abbas heads the internationally recognized Palestinian Authority in the Israeli-occupied West Bank and seeks a two-state solution with Israel. United States officials have suggested that the authority — a fierce rival of Hamas — could help govern Gaza after the war, an idea Mr. Netanyahu has rejected.

Complicating the picture, Mr. Erdogan told members of his party last week that before Mr. Haniyeh’s assassination, Turkey had been planning to invite him to address the Parliament too.

Despite the harsh rhetoric, Turkey has not cut diplomatic ties with Israel, although they have been scaled back and Turkey has announced the suspension of trade.

Turkey has also proposed that it serve as a “guarantor” of a Gaza cease-fire with Hamas in an arrangement under which the United States would secure Israel’s compliance. But that proposal has not gained traction.

Mr. Flake said that despite Mr. Haniyeh’s death, Turkey and the United States still shared the same goal.

“You can imagine that it has been difficult, but we’re both, our two countries, seeking the same thing: a cease-fire that will lead to some kind of enduring peace,” he said.

But the anti-Israel and pro-Hamas statements by Turkey’s leaders have made it hard for Turkey to play a central role in cease-fire negotiations as Qatar and Egypt have, Mr. Flake said.

“In terms of playing a mediating role, the rhetoric makes it very difficult,” he said.

Hezbollah, Iran’s most powerful regional proxy, has repeatedly said that only an end to Israel’s war in Gaza will lead it to cease its cross-border attacks.

“We continue to believe that a diplomatic resolution is achievable because we continue to believe that no one truly wants a full-scale war between Lebanon and Israel,” Mr. Hochstein told reporters after meeting with Lebanon’s Parliament speaker, Nabih Berri, an influential Hezbollah ally who passes messages back and forth between U.S. officials and the militant group.

Mr. Hochstein warned, however, that there was “no more time to waste and no more valid excuses from any party for any further delay.”

Mr. Hochstein was on his fifth trip to Lebanon since the outbreak of the conflict in Gaza more than 10 months ago, when Hezbollah began firing rockets into northern Israel in solidarity with Hamas. Since then, the fighting along the Lebanese-Israeli border has killed over a hundred civilians, displaced more than 160,000 people in both countries and threatened to expand into an all-out war.

The twin killings last month of top Hezbollah and Hamas leaders have intensified fears of a wider regional conflagration, with the region on edge awaiting the expected retaliation against Israel from Iran and Hezbollah.

“We have to take advantage of this window for diplomatic action and diplomatic solutions,” Mr. Hochstein said in Beirut on Wednesday. “That time is now.”

Mr. Ben-Gvir and a crowd estimated at about 2,000 inflamed tensions with leaders across the world and in Israel.

Prime Minister Benjamin Netanyahu of Israel assailed Mr. Ben-Gvir on Tuesday, in the latest sign of friction between members of the country’s fragile governing coalition.

“It is the government and the prime minister who determine policy on the Temple Mount,” Mr. Netanyahu’s office said in a statement, noting that there was no “individual policy” for any minister and that Mr. Ben-Gvir’s decision represented “a deviation from the status quo.”

The actions were taken around the world as a provocation, particularly given that diplomats have been scrambling to calm tensions in the Middle East and hoping that a cease-fire deal between Israel and Hamas would prevent a further escalation of the conflict following the assassinations last month of a Hezbollah commander in Lebanon and a Hamas leader in Iran. Israel has claimed responsibility for the death in Lebanon and is widely believed to have been behind the one in Iran. Both Iran and Hezbollah have vowed to retaliate.

In a briefing with reporters on Tuesday, Vedant Patel, a deputy spokesman for the State Department, called Mr. Ben-Gvir’s actions “unacceptable” and noted that the move “detracts” from efforts to reach a cease-fire agreement “at a vital time.”

Qatar, which has been among the nations mediating the negotiations between Israel and Hamas, condemned the prayers at the holy site as an attack “on millions of Muslims around the world.” It warned in a statement from its Foreign Ministry on Tuesday that the move could negatively affect the cease-fire talks.

Saudi Arabia’s Foreign Ministry also issued a statement condemning Mr. Ben-Gvir’s decision. It called the move “a provocation to the feelings of Muslims around the world, especially in light of the continuing war and acts of violence against defenseless Palestinians.”

Josep Borrell Fontelles, the European Union’s high commissioner for foreign affairs, also issued a statement “strongly” criticizing “the provocations” by Mr. Ben-Gvir. And France’s Foreign Ministry decried Mr. Ben-Gvir’s defiance of a “longstanding ban on Jewish prayer at the Al-Aqsa mosque,” urging Israel to respect the status quo. “This new provocation is unacceptable,” the French ministry said.

For years, the Israeli government has quietly allowed Jews to pray at the site, but in the videos from the scene on Tuesday, dozens of Jewish visitors are seen fully prostrating themselves in prayer. Some religious officials inside Israel expressed alarm at the flagrant violation.

Moshe Gafni, chair of the religious party United Torah Judaism, said Mr. Ben-Gvir was damaging the Jewish people and defying the dictates of generations of Israel’s chief rabbis. Michael Malchieli, Israel’s religious affairs minister and a member of the ultra-Orthodox Shas party, said Mr. Ben-Gvir’s actions were an “unnecessary and irresponsible provocation against the nations of the world.”

Mr. Ben-Gvir, a settler whose government responsibilities include oversight of the police, has not been circumspect about his expansionist aims or his opposition to a Palestinian state. He strongly opposes a cease-fire with Hamas, and his decision to lead a group to the sensitive site for prayers just as negotiations were set to resume underscored disagreements within Israel over the wisdom of striking a deal and halting the war in Gaza.

There are about 115 hostages — dead and living — believed to still be held in Gaza. Relatives of the hostages on Tuesday accused Mr. Ben-Gvir of repeatedly trying to thwart a cease-fire deal, saying the minister was endangering the chances of bringing their captive family members home.