President Obama, who publicly pushed for the United Kingdom to remain in the European Union, said Friday that the United States must “respect” British voters’ decision to exit.

“The people of the United Kingdom have spoken, and we respect their decision,” Mr. Obama said in a statement.

“The special relationship between the United States and the United Kingdom is enduring, and the United Kingdom’s membership in NATO remains a vital cornerstone of U.S. foreign, security, and economic policy,” he said.

The outcome of the Brexit vote came as a blow to Mr. Obama and British Prime Minister David Cameron, who announced he would step down in the fall to clear the way for “fresh leadership” for the exit.

It was also a blow to German Chancellor Angela Merkel, who called the Brexit vote “a cut for Europe.”

“Today is a cut for Europe. It is a cut in regards to European Unity,” Mrs. Merkel said in Germany, one of the wealthiest and most powerful countries in the EU. “We need to be calm and analyze the situation.”

Mr. Obama vowed that the U.S. relationship with the EU would also remain strong.

Mr. Obama praised the EU for doing “so much to promote stability, stimulate economic growth, and foster the spread of democratic values and ideals across the continent and beyond.”

“The United Kingdom and the European Union will remain indispensable partners of the United States even as they begin negotiating their ongoing relationship to ensure continued stability, security and prosperity for Europe, Great Britain and Northern Ireland, and the world,” he said.

Vice President Joseph R. Biden briefly addressed the “Brexit” in a speech Friday across the Irish Sea in Dublin, saying the U.K. will remain an “indispensable” partner in assuring America’s economic and national security, even though the administration urged the kingdom to stick with Europe.

“Not how we would have preferred it to be, but respect their position,” he said in a wide-ranging speech on the Irish-American experience, the global economy and what he viewed to be the dangerous rise of xenophobic politics in America and Europe.

Mr. Biden said the U.S. will give EU leadership in Brussels the breathing room to figure out “what this new relationship will look like.”

• Tom Howell Jr. contributed to this article.

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