At the same time, Mr. Obama has had an ambivalent relationship with Europe during his presidency. His heavy emphasis on Asia — a policy dubbed the pivot — stoked suspicions in Europe that he was moving away from the continent to the faster-growing markets of the East. In his first term, the centerpiece of his Europe policy was an effort to “reset” relations with Russia.

Critics said the tendency to take Europe for granted predated Mr. Obama. “Since 2000, both the Bush and Obama administrations have acted as if Europe as a task had been solved and that we no longer needed to ‘tend the garden’ as George P. Shultz used to say,” said John C. Kornblum, a former American ambassador to Germany, referring to Ronald Reagan’s secretary of state. “The Europeans played their part by acting as if they didn’t need us.”

Even after Mr. Obama worked closely with Europeans on difficult issues like the NATO air campaign in Libya, there was a sense that he looked on the trans-Atlantic alliance with a gimlet eye. In April, he struck a nerve by suggesting that Britain and France had been “free riders” in that operation, leaving the United States to bear most of the military burden.

Some critics suggest Mr. Obama’s reluctance to be more militarily involved in Syria had an indirect effect on the British vote because of the flood of refugees the civil war has sent to Europe, destabilizing the continent and firing up nativist sentiment. Syrian refugees, however, account for far less of Britain’s immigrant population than they do in Germany, for example.

Mr. Obama has a chance to demonstrate his support for Europe at a NATO summit in Warsaw next month. But there again, the loss of Britain as a member of the European Union will be felt. Britain has historically been one of NATO’s strongest boosters. It has resisted initiatives like a joint European military headquarters because it could compete with NATO. European officials said Germany and France might revive the proposal as a way to reinforce Europe’s unity in the wake of the British vote.

Britain’s decision to leave Europe just as Mr. Obama was putting on an extravagant celebration of entrepreneurship and engagement in Silicon Valley undercut his message that innovation, open borders and free trade can improve people’s lives. It is the same assertion that has also underpinned his efforts to forge a new dynamic in the Middle East.