The Obama administration is engaged in tortuous negotiations with Turkey about the scale and nature of its participation in the military campaign against the Islamic State. And while Mr. Obama lined up Saudi Arabia, the United Arab Emirates, Jordan and Qatar to take part in airstrikes against the militants in Syria, preserving the coalition over a prolonged campaign that could require Arab ground troops will be challenging.

“The fact that he called the leaders of these, or at least senior officials, in both of these countries to apologize is an indication that he himself wishes he had said it a little bit differently,” Mr. Earnest said.

Mr. Biden’s gaffe overshadowed what was meant as a major foreign policy address, in which the vice president sought to put the multiple upheavals in the world — the Islamic State, the Ebola outbreak and the confrontation with Russia over Ukraine — into a broader context.

The vice president just hired a new national security adviser, Colin H. Kahl, a former Pentagon official and a Georgetown scholar who specializes in Iran. Mr. Kahl, who worked as a foreign policy adviser to the 2012 Obama campaign, had a major hand in the speech.

Several officials said Mr. Biden’s contrite phone calls to President Recep Tayyip Erdogan of Turkey and the crown prince of Abu Dhabi, Mohammed bin Zayed, had cleared the air. Mr. Biden takes particular pride in his relationship with the prickly Mr. Erdogan.

In December 2011, he was granted a two-hour session with Mr. Erdogan, then the prime minister, when he was recuperating at his private residence from a medical procedure. Mr. Biden and his aides put on slippers and were introduced by Mr. Erdogan to his son, daughter and son-in-law.

In a recent conversation, Mr. Biden said, the Turkish leader confided, “You were right — we let too many people through” and was now trying to seal its border. Mr. Erdogan denied saying that and told reporters, “If Mr. Biden has said such a thing at Harvard, he needs to apologize.”