In the YouTube video, Mr. Cuomo said his conservative critics had frequently used the name as an insult against him.
On Tuesday morning, President Trump, who often criticizes the Cuomo brothers, seemed to drive that point home, writing on Twitter: “I thought Chris was Fredo also. The truth hurts. Totally lost it.”
Mr. Trump’s eldest son, Donald Trump Jr., also weighed in on Twitter. Addressing Mr. Cuomo on Monday night, he wrote: “Take it from me, ‘Fredo’ isn’t the N word for Italians, it just means you’re the dumb brother.”
The Cuomo family has long chafed at even the hint of a Mafia reference. Mario Cuomo frequently urged reporters and public officials not to use the word “Mafia” — even when a mob boss was killed in 1985 — calling it an “ugly stereotype.”
Brandon Recor, the host of the YouTube channel that posted the video, said he was sent the footage on Sunday night by the man involved in the confrontation, who Mr. Recor said requested anonymity because he lived in New York and was worried about retribution from the governor.
Mr. Recor, who lives in Los Angeles, said the man was at a bar in Shelter Island, N.Y., when he approached Mr. Cuomo. In the video, the man insists he thought Mr. Cuomo’s name really was Fredo.
Mr. Recor claimed that “the guy listens to Rush Limbaugh a lot, and for whatever reason, Rush Limbaugh refers to Chris Cuomo as Fredo. So the guy was like, ‘Hey, Fredo, can I get a photo?’”