“His timing is pretty good,” said David Axelrod, a former senior adviser to Mr. Obama. “If he arrived the week before, the party would have been rip-roaring. Now everybody is sitting there with an ice pack on their forehead. They might be ready to listen to some sensible advice.”

The departure of Britain from the European Union is likely to have significant, if not immediate, effects on Europe’s security. Some experts express fear that it will weaken the response to Russian aggression in Ukraine, for example. At the Warsaw meeting, a senior administration official said, Mr. Obama plans to emphasize the need for the European Union to cooperate more closely with NATO.

This has long been a goal of the United States, but the exit of Britain from the union makes it more urgent, said the official, who spoke on the condition of anonymity in advance of the NATO meeting. NATO and the European Union could collaborate in countering cyberattacks, coordinating military exercises and patrolling the central Mediterranean.

But Britain’s departure could pose problems for NATO. European diplomats said they expected France and Germany to consider closer military integration within the European Union — something Britain and the United States have resisted because they view it as competing with the alliance.

Mr. Obama spoke out publicly against a “leave” vote when he visited Britain in April, and the outcome of that debate was perhaps a harbinger of the challenges he faces in resisting antiglobalism at home. In his speeches, he emphasizes that leaders must recognize the fears and resentments that people feel because of economic dislocation.

“For them, globalization is inherently rigged toward the top 1 percent,” Mr. Obama said in an address to the Canadian Parliament. “I understand that vision. I know why it’s tempting.”

The problem, the president said, is that it is too late to turn back.

“Restricting trade or giving in to protectionism in this 21st-century economy will not work,” Mr. Obama said. “Even if we wanted to, we can’t seal ourselves off from the rest of the world. The day after Brexit, people looked around and said: ‘Oh! How is this going to work?’”