“This one was really about America’s role in the Pacific, and I think pulling out was a signal we’re not as interested,” said William Reinsch, the Scholl Chair in International Business at the Center for Strategic and International Studies and a former Clinton administration official. “The president can say whatever he wants, but the way this is being perceived is as a pullback of American influence.”

The Trump administration has pushed back against claims that the America First doctrine is isolationist and it is ceding global leadership. While advisers have continued to criticize global institutions, they insist their goal is to improve, not destroy, them.

“We’re going to the World Economic Forum to share President Trump’s economic story and to tell the world that America is open for business,” Gary D. Cohn, who heads the White House National Economic Council, said in a briefing on Tuesday. “America First is not America alone.”

Mr. Trump and his advisers say that the United States will be pushing ahead with new trade deals — ones that will ultimately be better for American companies and their workers.

“We’ll be doing other trade deals,” Mr. Trump said Monday as he signed an order imposing tariffs on imports of washing machines and solar modules. “We’re in the process of negotiating with other countries, also, all of which have treated us very unfairly.”

Yet the willingness of countries to engage with the United States is unclear. For many, a relationship with the Trump administration has been a delicate dance: They do not want to risk access to the American market, or raise the ire of the American president.

But global leaders are warily watching the Trump administration’s take-it-or-leave-it approach to renegotiating its trade pacts with Canada, Mexico and South Korea. They have seen Mr. Trump scrap the global climate change accord, withdraw the United States from the United Nations compact on migration and refugees and criticize global institutions that the United States largely designed, like the United Nations, the World Trade Organization and the North Atlantic Treaty Organization.