After Mr. Carlson posted the article on his Web site, timed for the prime-time Fox News programming lineup, he appeared on the Sean Hannity program on Fox to explain what he had found. More than three million people tuned in, a substantially larger audience than usual. There was the president, speaking in a way that he usually does not in public, telling a black audience, a group of clergy members at Hampton University in Virginia, that the government did not care about them.

The president was using racial tensions to try to divide America into different classes of people, Mr. Carlson argued. And the accent? To him, it was further evidence of the argument that many Obama opponents on the right have been pushing in their writings, talk shows and films for years: We don’t really know who this man is.

A conspiracy theory cottage industry has sprung up around the notion that Mr. Obama is somehow foreign, if not by birth than by ideology. Donald Trump breathed new life into a career as a cable news pundit by repeatedly questioning if the president’s birth certificate was authentic.

Dinesh D’Souza, the conservative author and filmmaker, has a new movie in theatrical release called “Obama 2016” that argues that Mr. Obama’s father, a Kenyan, instilled anti-Americanism in his son at an early age.

One of the film’s financial backers was Mr. Ricketts, the founder of TD Ameritrade, who considered getting behind a multimillion-dollar ad campaign that would have linked Mr. Obama to his former pastor, Mr. Wright, who became a source of embarrassment for the president.

In many ways, Mr. Hannity was an ideal first stop for Mr. Carlson. Throughout the year Mr. Hannity has featured a segment called “Vetting the President,” often focusing on foibles from Mr. Obama’s past or over his tenure. As Mr. Hannity said in March, “We call it ‘Vetting the President.’ Because the mainstream media, they’re not going to do it. They helped elect him. They hid a lot of things about his past.”

And after the video’s release on The Daily Caller and Mr. Hannity’s program, it was the talk of the rest of the conservative news media. “Clearly race-baiting, clearly angry, and I’m telling you: This is who he is to this day,” Rush Limbaugh told his radio audience on Wednesday. Mr. Carlson declined to say on Tuesday how he acquired the video, which news networks have had in their libraries since it was shot. He said only that he had received the video in the last few days.