Israel also sought to play down the impact of the expulsion of its ambassador, saying Mr. Levy had already completed his tour of duty and had been about to leave for home.

Turkey once ranked as Israel’s closest strategic ally in the Muslim world, but ties began to fray with an Israeli military operation in Gaza in late 2008 and early 2009.

The most recent deterioration in relations with Turkey has compounded Israel’s regional woes from the Arab Spring, which has redrawn the geopolitical landscape along Israel’s borders. Earlier this year, the revolution in Egypt ousted President Hosni Mubarak , once a crucial ally, while the bloody and unpredictable crackdown on dissent in Syria has raised yet more questions about regional stability.

Mr. Davutoglu said Friday that Israel had failed to grasp the consequences of “gigantic changes in the Middle East region.”

Turkey’s move on Friday came as a long-awaited United Nations review of the 2010 raid on the flotilla found that, while Israel’s naval blockade of Gaza was both legal and appropriate, the way Israeli forces boarded the vessels trying to break that blockade 15 months ago was excessive and unreasonable.

The report also found that when Israeli commandos boarded the main ship, they faced “organized and violent resistance from a group of passengers” and were therefore required to use force for their own protection. But the report called the force “excessive and unreasonable,” saying that the loss of life was unacceptable and that the Israeli military’s later treatment of passengers was abusive. Eight Turks and an American of Turkish descent died in the Israeli raid.

A copy of the 105-page report was obtained early by The New York Times.

In a statement on Friday, Israel’s prime minister’s office said that “as any other state, Israel has the right to defend its civilians and soldiers.”

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Israel’s refusal to apologize and Turkey’s insistence on an apology appeared to further cement the impasse. But both sides seemed ready on Friday to at least display readiness for further efforts to overcome it.

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“Israel cherishes the significant ties, past and present, between the Turkish and Jewish peoples,” the Israeli statement said. “For that reason, the State of Israel has made numerous attempts in the last few months to settle the dispute between the two countries, but regrettably, these attempts have not been successful. The State of Israel hopes that a way will be found to move beyond this discord and will continue its endeavors to that end.”

Privately, however, Israeli officials noted that relations between the former allies had started deteriorating long before the flotilla raid. They questioned whether those relations would be fully restored under any circumstances given what some here consider a steady reorientation in Turkish foreign policy away from Israel, in the direction of the Arab world and Iran .

An apology might bring the relationship back to how it was on the eve of the raid, said one official, speaking on condition of anonymity because of the delicate diplomatic situation. But, he said, “It will not put us back to how things were three years ago.”

Dore Gold, a former Israeli ambassador to the United Nations and president of The Jerusalem Center for Public Affairs, a conservative-leaning research institute, said that “every Israeli official would like to see a restoration of Israel-Turkey relations in the longer term.” The question, he said, was whether the crisis was primarily over the flotilla raid, or whether that episode had “only put into sharper relief a more fundamental change.”

Israeli officials said that while Turkey’s expulsion of the Israeli ambassador marked a sharp diplomatic downturn in relations with Israel, military ties had already been undergoing serious erosion since 2009. “My feeling is that there is not a lot to suspend in that sense,” one official said.

In October 2009, Turkey asked Israel not to participate in a multinational air force exercise that was supposed to take place in Turkey. In June 2010, after the raid on the protest ship, Turkish officials said that Turkey had been blocking Israeli military flights from entering its airspace.

Turkey has been an important customer for Israel’s defense industries in the past, and an important partner for military and intelligence cooperation at many levels.

There has been growing debate in Israel about whether to apologize. Some senior officials have advised against it, saying that an apology denotes an acceptance of responsibility for what happened. Some have also pointed to the importance of upholding Israeli morale. Israel’s ultranationalist foreign minister, Avigdor Lieberman , said in July that an apology would be “humiliating” for Israel and tantamount to abandoning its soldiers.

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Others say that given Israel’s regional isolation and the strategic interests that Israel shares with Turkey, it is a price that the country has to pay.

“Just look at the map,” said Shlomo Brom, a retired general now at the Institute for National Security Studies at Tel Aviv University. “Turkey is located right by Syria, Iraq and Iran.”

He asked: “What is so hard about apologizing? Undoubtedly mistakes were made and people were killed.”