Many conspiracy theorists and far-right commentators are spreading doubts that President Biden is alive after he posted a letter on Sunday resigning from his presidential campaign.

The theory is baseless. Mr. Biden, who learned he had Covid on July 17 and has been in isolation since, called into a campaign event for Vice President Kamala Harris on Monday and the two exchanged praise in a playful banter.

“Joe, are you watching? You hear this clapping? Can you see it?” Ms. Harris asked the president. As the room echoed with applause, Mr. Biden responded, “I’m watching. I’m watching it.”

Mr. Biden has also said he would address the nation later this week.

Despite all that, the conspiracy theories have nevertheless gained traction among the far-right, earning support from some mainstream commentators. A hashtag on the topic became a Top 10 trending topic in the United States on X.

The theory adds to a torrent of falsehoods and misleading posts by right-wing influencers that have spread since Mr. Biden’s announcement. They have falsely denigrated Democrats for conducting a “coup” against primary voters by removing Mr. Biden and have attacked Ms. Harris with a deluge of sexist and racist remarks.

Soon after the letter was posted to social media on Sunday, Fox News anchors raised questions over the lack of photos or videos to accompany the announcement.

“It is disconcerting, isn’t it?” Bret Baier, a Fox News host, said.

“Proof of life, please,” added Dana Perino, a co-host.

Online, conspiracy theorists raced to analyze the letter for signs that something was amiss. Several had fixated on Mr. Biden’s autograph, which clearly bears the name “J.R. Biden” but also contains an underline not seen in other signatures from the president.

Bill Ackman, the billionaire financier, promoted several posts on X questioning whether the signature was authentic.

“If this were a hostage situation, that letter would not qualify as proof of life,” he wrote on X in a post that received more than 1 million views.

Several commentators also noted that the lack of a White House seal or letterhead was evidence that the document was not actually written by Mr. Biden. His candidacy is separate from his duties at the White House or for the federal government.

As the conspiracy theory circulated on Monday — and Mr. Biden remained out of the limelight as he recovers — the idea gained support from more prominent right-wing influencers.

Charlie Kirk, the right-wing podcaster, shared a convoluted story that suggested Mr. Biden was “dying or possibly already dead.” It was seen more than 12 million times on X.