“Maybe there was some sort of presumption on the part of the president and his team that if the U.S. said stop, this process would come to a halt,” said Phil Levy, a senior fellow at the Chicago Council on Global Affairs and an economist in the George W. Bush administration. “What this shows is that’s not true. The world just moves on without us.”

In July, Japan signed a wide-ranging new trade deal with the European Union — a step the Japanese prime minister, Shinzo Abe, referred to as hoisting “the flag of free trade high amidst protectionist trends.” The European Union pushed ahead with a major update to its agreement with Mexico, while China pursued a pan-Asian agreement, among other deals.

Business interests in the United States are watching with alarm as other countries strike agreements that exclude American exporters. For example, ranchers in Canada and Australia will be able to sell beef at lower prices in Japan than their American competitors, who will be subject to higher tariffs because the United States is not party to the Trans-Pacific Partnership.

Kent Bacus, the director of international trade and market access for the National Cattlemen’s Beef Association, said Tuesday that the United States’ withdrawal was “a missed opportunity for the United States to gain greater access to some of the world’s most vibrant and growing markets.”

Chancellor Angela Merkel of Germany during her address on Wednesday at the World Economic Forum in Davos, Switzerland. Credit Markus Schreiber/Associated Press

“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.

In Davos, at a forum long considered the center for globalization, the administration’s America First message has not been entirely well received. On Wednesday, comments by Wilbur Ross, the commerce secretary, that American troops were “coming to the ramparts” in what he called a continuing global trade war prompted blowback from other state leaders.

Chancellor Angela Merkel of Germany warned on Wednesday that “protectionism was not the answer” and called right-wing populism a “poison,” in a speech largely seen as a rebuttal to Mr. Trump before his expected speech on Friday. “We think that shutting ourselves off, isolating ourselves, will not lead us into a good future,” Ms. Merkel said.

Canada has also contrasted its approach with that of the United States. Chrystia Freeland, the Canadian foreign minister, said she was worried about movements around the world that have blamed trade and immigration for the hollowing out of the middle class, and that Canada was “doubling down” on its openness to trade and immigrants. “It’s partly a statement about our values. But we also think it is a really great economic benefit to us as well,” she said.

So far, it is unclear which countries the United States could be courting to create new trade deals. Britain has expressed interest in a bilateral deal with United States, but talks could not begin until it finishes extricating itself from the European Union, which does not look likely to be resolved soon.

Japan, the preferred partner for many American businesses, appears to have spurned the United States’ offer to forge a trade deal one on one, after Mr. Trump pulled the United States out of the Trans-Pacific Partnership.

And China has tried to position itself as a global leader — a sharp change from the early 1990s, when the leader at the time, Deng Xiaoping, cautioned his countrymen to “hide your strength and bide your time.”

China has set about building its own vision of an international order, including an Asian investment bank to rival the World Bank, and its global infrastructure project, One Belt One Road. At last year’s meeting in Davos, President Xi Jinping of China portrayed his country as a global leader and vigorously defended free trade — a position some considered ironic for a country that maintains sharp controls on many industries.

The Trump administration’s more aggressive stance toward allies and adversaries may shift this balance in China’s favor. At an event on Tuesday in Washington, Juan Gabriel Valdés, the Chilean ambassador to the United States, contrasted Chile’s now warm relationship with China with the United States’ more combative stance on trade, especially in its negotiations over Nafta.

“In the history of the region, we have never had a partner that is so willing, so open, so dedicated to listen to what are our needs than what we have now with China,” Mr. Valdés said.

“The United States continues to be, and will probably be in the next 10, 20 years, the main partner of the Latin American region,” he said. “But change is there. It’s written on the wall, if this continues like it is.”