SEOUL — President Biden embarked Thursday on his first diplomatic mission to Asia since taking office, hoping to demonstrate that the United States remained focused on countering China, even as his administration stage-managed a war against Russia in Europe.

With his original strategy of pivoting foreign policy attention to Asia effectively blown up by Russia’s invasion of Ukraine, Mr. Biden has now shifted to the argument that there can be no trade-off between Europe and Asia and that only the United States can bring together the democracies of the East and West to stand up to autocracy and aggression in both spheres.

For Mr. Biden, finding his balance between the twin imperatives will require geopolitical maneuvering that would challenge any president. The competing demands on his time and attention were on display on Thursday as he squeezed in a last-minute meeting at the White House with the leaders of Sweden and Finland to welcome their decisions to join NATO before heading to Joint Base Andrews to board Air Force One for the long flight to South Korea. And days before that, Mr. Biden hosted Southeast Asian nations at the White House to detail new investments in clean energy and maritime assets, part of an effort to prevent China from dominating the Indo-Pacific.

“What the administration is trying to do is add credibly to their claim that America is back as a global leader and the idea that the world is not two theaters,” said Evan S. Medeiros, a scholar at Georgetown University who served as an Asia adviser to President Barack Obama when the original pivot to Asia was conceived. “It’s, ‘Hey, I’m not going to forget about you; this is not a choice between Europe and Asia.’”

The Ukraine war will no doubt follow Mr. Biden during stops in Seoul and Tokyo, hovering over his talks with the leaders of South Korea, Japan, Australia, India and others. At the same time, administration officials fear that North Korea may use the president’s trip to thrust itself back onto the global agenda with an in-your-face test of a nuclear weapon or intercontinental ballistic missile, reminding everyone of dangers beyond Ukraine.

“We are preparing for all contingencies, including the possibility that such a provocation would occur while we are in Korea or in Japan,” Jake Sullivan, the president’s national security adviser, told reporters this week before Mr. Biden departed Washington. Mr. Sullivan has consulted his counterpart in China in recent days to discuss, among other things, the prospect of a North Korean provocation.

Mr. Biden’s trip is also aimed at reassuring allies in the region who were rattled by President Donald J. Trump’s unorthodox approach to Asia in recent years. Mr. Trump pulled the United States out of the Trans-Pacific Partnership, an American-negotiated regional trade pact meant to counter China’s growing economic sway. He repeatedly questioned American troop commitments to South Korea and the mutual defense agreement with Japan, while engaging in what he called a “love affair” with North Korea’s Kim Jong-un.

Bruce Klingner, a longtime C.I.A. analyst on Asia now at the Heritage Foundation, said South Korea and Japan were increasingly nervous about North Korea’s capabilities and Mr. Trump’s threats to pull back from the region. “Biden should provide unequivocal assurances of U.S. dedication to the defense of our allies and affirm the U.S. extended deterrence guarantee of nuclear, conventional and missile defense forces,” he said.

A couple of recent studies have concluded that even though American political influence in the region has risen again with Mr. Trump leaving power, the United States has continued to lose economic influence because of the withdrawal from the Trans-Pacific Partnership.

“The biggest criticism of the administration in Asia right now is they have no economic strategy and they’re ceding the field to China,” said Michael J. Green, the incoming chief executive of the United States Studies Centre in Australia and a former Asia adviser to President George W. Bush.

To address that, Mr. Biden plans to unveil a new Indo-Pacific Economic Framework, which is a pale shadow of a full-scale trade pact but will outline various mutual priorities like digital trade and supply chain security. American officials hope it will be joined by many of the countries still in the Trans-Pacific Partnership.

Mr. Green called that an important first step but one that behind the scenes, the Japanese, Australians and others find inadequate for the moment — although they are unlikely to say so publicly. “A lot of their interest is to show the U.S. is back and China is not going to write the economic rules,” Mr. Green said.

Matthew P. Goodman, senior vice president for economics at the Center for Strategic and International Studies, said that if the Biden administration did not offer greater access to the American market, nations in the region would be looking for direct funding to expand infrastructure and the digital economy. “I think a lot of partners are going to look at that list and say: That’s a good list of issues. I’m happy to be involved,” said Mr. Goodman. “But, you know, are we going to get any tangible benefits out of participating in this framework?”

In crafting the economic framework, Biden administration officials have focused in part on labor and environmental standards. But without the benefits of lowered trade barriers, other countries may be reluctant to make costly commitments.

“The bottom line is the United States is not coming to the table with market access,” said Sheila A. Smith, a senior fellow for Asia-Pacific studies at the Council on Foreign Relations. “And that’s the trade piece. That’s what the region is looking for.”

During stops in Seoul and Tokyo, Mr. Biden will encounter two new partners who are both seen as more aligned with American priorities and likely to have good chemistry with the president, according to Mr. Green and other analysts and officials. The first, President Yoon Suk-yeol of South Korea, was inaugurated on May 10 and has taken a stronger approach to China and North Korea than his predecessor, while the second, Prime Minister Fumio Kishida of Japan, was elected just last October and enjoys a level of popularity that is likely to keep him in office for the duration of Mr. Biden’s term, unlike the frequent revolving-door governments in Tokyo.

“Inevitably North Korea is going to make itself front and center as part of the agenda for a Biden-Yoon summit,” said Scott A. Snyder, the director of U.S.-Korea policy at the Council on Foreign Relations. “Just the fact that that speculation is out there makes it necessary for the two leaders to talk about extended deterrence, how that works, and to try to deepen their shared commitment to security and defense.”

While in Tokyo, Mr. Biden will also meet with other leaders of the so-called Quad — the United States, Japan, Australia and India — his second time sitting down with his counterparts in a bloc meant to resist Chinese hegemony in the region.

With Australian elections scheduled for Saturday, it remained unclear who would attend the meeting, on Tuesday.

But the most complicating factor may be how Mr. Biden approaches Prime Minister Narendra Modi of India, who has been hesitant to condemn Russia’s invasion of Ukraine out of fear of undermining security ties with Moscow. Moreover, Mr. Biden’s promise to combat autocracies around the world will be put to the test with Mr. Modi, who has marginalized and maligned minority Muslims.

But the president’s aides said he can press the international campaign to thwart Russian aggression while still navigating the diplomatic complexities of the Asian-Pacific region and reaffirming America’s role in this part of the world.

“He remains focused on ensuring that our efforts in those missions are successful,” said Mr. Sullivan, “but he also intends to seize this moment, this pivotal moment, to assert bold and confident American leadership in another vital region of the world.”