The Guardian Angels founder Curtis Sliwa was being interviewed live from Times Square on Tuesday night by the Fox News host Sean Hannity when their exchange took a startling turn.

The topic was what both men seemed to agree was a migrant-fueled wave of crime and chaos that they claimed had overtaken New York amid a surge of arrivals into the city from the southern border over the past two years.

Suddenly, Mr. Sliwa had a prime-time example. As he spoke, the half-dozen red-jacketed Angels flanking him slipped out of the frame.

“Our guys have just taken down one of the migrant guys right here on the corner of 42nd and Seventh where all this is taking place,” Mr. Sliwa said, pointing off camera.

“Can you pan the camera?” Mr. Hannity asked his cameraman.

The cameraman swung around and captured the Angels confronting a slightly built man in a hooded sweatshirt, throwing him to the ground and putting him in a headlock.

“He is out of control,” Mr. Sliwa said as the camera turned away and Mr. Hannity shifted to criticizing President Biden over his administration’s border policies.

On Wednesday, however, the police said the man was not a migrant but a resident of the Bronx. They also said he had been issued a summons for disorderly conduct after trying to disrupt the interview and for being “loud, disorderly and threatening on a crowded sidewalk.” He was not charged with shoplifting, as Mr. Sliwa had suggested on air that he would be.

The episode came in the wake of two high-profile criminal cases that have provided fodder for those critical of the migrants’ effect on the city: the arrests of about a half-dozen people in a robbery spree and an attack on two police officers in a Times Square brawl.

Although Mayor Eric Adams and other administration officials have been clear in saying that the vast majority of migrants coming to New York follow the law, the mayor has also said repeatedly that the influx will “destroy” the city, and the police commissioner, Edward A. Caban, said in announcing the robbery arrests that “a wave of migrant crime has washed over our city.”

Murad Awawdeh, the executive director of the New York Immigration Coalition, said on Wednesday that the rhetoric being used by the mayor and others had emboldened a “vigilante” like Mr. Sliwa to take the law into his own hands in ways that threatened all New Yorkers.

“This is what happens when Eric Adams and the Republicans go to extreme lengths to scapegoat immigrants for every problem they have been unable to solve,” Mr. Awawdeh said.

“It needs to stop,” he added.

A spokesman for Mr. Adams declined to comment on the episode involving Mr. Sliwa and said that New York City had provided shelter and care for more than 175,000 migrants while Republicans in Congress continued to block comprehensive immigration reform.

Asked about airing the confrontation live during Mr. Hannity’s show — and Mr. Sliwa’s claim, without proof, that the man being restrained was a migrant — Fox News said in a statement that “the situation took place during a live television broadcast” and that the network had “updated its audience as more information became available from the N.Y.P.D.”

The Guardian Angels have been a presence in New York since the late 1970s when Mr. Sliwa founded the group to patrol the streets and the subway during an era of rampant crime. Mr. Sliwa admitted years ago that he had fabricated some of the group’s early crime-fighting efforts to garner publicity and raise its profile.

After cutting away from the confrontation Tuesday night, Mr. Hannity returned to Mr. Sliwa later in the broadcast for an explanation of what the audience had seen.

“He had been shoplifting first, the Guardian Angels spotted him, stopped him — he resisted, and let’s just say we gave him a little pain compliance,” Mr. Sliwa said. “His mother back in Venezuela felt the vibrations.”

He added: “We’ve got to take 42nd Street back, Sean, these illegals think they own this street. They think they rule the night. This is our country.”

In an interview on Wednesday, Mr. Sliwa, who, as the Republican nominee, lost the 2021 election to Mr. Adams, said he had merely repeated what he had heard from people in the crowd.

He also said the Guardian Angels had confronted the man after he assaulted a female member of the group. It was after the initial encounter, he said, that people who gathered nearby had said the man had been stealing items from stores in the area. A backpack found by the police, Mr. Sliwa said, contained about 10 items of children’s clothing still bearing anti-theft tags.

Asked why he would tell Mr. Hannity with such certainty what the man’s background was, Mr. Sliwa said again that he had simply passed along what he had been told.

“Is it possible he wasn’t a migrant?” he said. “Of course.”