In three days of diplomacy in Asia, President Biden rallied world leaders to help finance poor nations, fortified the coalition backing Ukraine and struck a deal with Vietnam to counter Chinese aggression.

But even before he left Vietnam on Sunday night, the president was hammered with a very different narrative.

By Monday morning, as the 80-year-old president was flying home on Air Force One, conservative media outlets had seized on his end-of-trip news conference as the latest evidence that he is too old to perform on the world stage.

It is a pattern that infuriates the White House, where Mr. Biden’s top aides believe that stories about the president’s age and health are stoked by his enemies in an effort to undermine his accomplishments.

In their coverage of the Hanoi news conference, conservatives took aim at Mr. Biden. The Daily Mail, which frequently mocks the president, was typical: “A sleepy President Joe Biden saw his rambling Vietnam press conference brought to a sudden end on Sunday night with his mic cut and jazz music playing him off the stage, like he went too long in an awards speech,” the site wrote.

Although Mr. Biden fielded questions on a range of foreign policy issues, including China’s economic troubles and climate change, he spoke softly and appeared tired. To make a point about climate deniers, Mr. Biden rambled into a familiar story about liars in a John Wayne movie, which left some in the audience deeply perplexed.

“There’s a movie about John Wayne. He’s an Indian scout. And they’re trying to get the Ap- — I think it was the Apache — one of the great tribes of America back on the reservation,” he said, continuing to discuss “Indians in headdresses,” “Union soldiers” and a quote from an Indian to John Wayne’s character.

“The Indian looks at John Wayne and points to the Union soldier and says, ‘He’s a lying, dog-faced pony soldier,’” Mr. Biden said. “Well, there’s a lot of lying, dog-faced pony soldiers out there about global warming.”

The president also struggled to read from notecards that listed the names of reporters to call on for questions.

It was far from the first time the president’s rivals have used his words and actions to undercut him, especially on social media platforms that do not emphasize the context or background for an event.

To Mr. Biden’s advisers, the political damage is coming not from legitimate questions about the president’s health or stamina, but from what one adviser once described as fake and distorted internet posts.

When Mr. Biden visited Maui to tour the area devastated by fire, for example, a camera caught him with his eyes cast down. Social media lit up, with conservatives pushing the idea that he had fallen asleep, even though the full video of the event showed that he had not.

White House aides recall the time conservative critics on social media fueled coverage of Mr. Biden’s fall off a bike near his Delaware beach house in the summer of 2022. (A minor fall that could happen to anyone, his advisers said.) Or the time that he tripped over a sandbag while handing out diplomas at the Air Force Academy commencement this year. (He got up quickly and was fine, they said.)

Andrew Bates, a White House spokesman, blamed the most recent headlines on conspiracy-minded critics.

“Since President Biden’s experienced leadership and drive are what transformed infrastructure from a punchline into a reality and Medicare drug negotiations from an abandoned promise to a 30-year breakthrough for patients, it actually makes perfect sense that the QAnon crowd keeps inadvertently admitting the best it can do is either keep moving their credibility into the red or complain that he likes dogs,” Mr. Bates said.

Then he echoed one of the president’s most puzzling jabs: “Best of luck in their senior year.”

Still, for any president, the performance at the news conference in Vietnam would not have ranked among the best.

Former President Donald J. Trump routinely delivered rambling, often nonsensical speeches and news conferences. He ranted about eagles killed by windmills and told fake stories about the U.S. military running out of ammunition. Often, Mr. Trump’s anger-filled diatribes made little sense.

By that standard, Mr. Biden was more capable.

But the news conference underscored what survey after survey suggests is his biggest political vulnerability: the perception that his age has eroded his ability to do the job.

A recent poll by The Wall Street Journal found that 73 percent of Americans believe the president is too old to seek a second term in the White House. In a Reuters/Ipsos poll in April, 73 percent said Mr. Biden was too old to be in office, compared with 53 percent who said the same about Mr. Trump, 77, his leading Republican opponent.

White House aides, and their counterparts in his re-election campaign, insist that the polls do not worry them. They say they are little more than a snapshot in time taken before voters are presented with a stark choice to make — such as another one between Mr. Biden and Mr. Trump.