Fauci, asked about it on CNN on Tuesday, said Watters should be fired "on the spot" but predicted he wouldn't be held accountable for his language.
Fox said Watters' words had been "twisted completely out of context".
Watters, a host on Fox News Channel's panel show "The Five" who made his initial mark doing aggressive interviews for Bill O'Reilly, spoke on Monday to a group of college and high school conservatives. His audience booed at the mention of Fauci's name.
Fauci, director of the National Institute of Allergy and Infectious Diseases and the government's most visible spokesman on the COVID-19 pandemic, has been the subject of frequent criticism by some Fox News commentators who have been seeking to appeal to audience members resistant to vaccinations.
Watters said that Fauci should be confronted on the subject of whether the National Institute of Health funded research at a lab in Wuhan, China, the city where the COVID-19 virus originated. He said an interviewer should suggest he lied about the topic — something Fauci has disputed.
"Now you go in for the kill shot, the kill shot with an ambush, deadly, because he doesn't see it coming," Watters said.
He suggested an interviewer say, "You know why people don't trust you, don't you?' Oh, he is dead. He's dead. He's done."
The interviewer should make sure the encounter is filmed and the footage given to conservative media, Watters said. It's a confrontation technique that has been used elsewhere in conservative politics by the group Project Veritas.
"Just make sure it's legal," Watters said.
A partial clip of Watters' speech, beginning with the "kill shot" quote, spread around the internet, with some commentators suggesting that he had advocated assassinating Fauci.
During an interview with Fauci on Tuesday, CNN's John Berman referred to Watters as a "Fox News entertainer," and asked about the comments without playing the clip, saying it was dangerous. Berman referred to a "rhetorical kill shot," and asked Fauci how much that language concerned him.
Fauci noted that for two years, he's been encouraging people to protect themselves against COVID-19 by practising good public health practices and get vaccinated.
"For that, you have some guy out there saying people should be giving me a kill shot, to ambush me?" he said.
"I mean, what kind of craziness is there in society these days? That's awful what he said. And he's going to go, very likely, unaccountable. Whatever network he is on is not going to do anything. The guy should be fired on the spot."
Fauci, in a "60 Minutes" interview in October, discussed death threats he had received and his need for a security detail.
In a statement, Fox said "based on watching the full clip and reading the entire transcript, it's more than clear that Jesse Watters was using a metaphor for asking hard-hitting questions ... and his words have been twisted completely out of context."
Watters' reference to "kill shot," however, baffled some people in television news.
"I've never used, or heard, that term used and I did my share of ambush interviews as an investigative producer," said Mark Lukasiewicz, dean of Hofstra University's School of Communication and a longtime journalist at NBC News.
Last month on Fox, Tucker Carlson compared Fauci to Italian World War II dictator Benito Mussolini, while Lara Logan said Fauci, to some people, represented Josef Mengele, the Nazi death doctor.
Fox has not commented on the statements that Carlson and Logan made about Fauci. Logan, a contributor to the Fox Nation streaming service who had appeared as a commentator on the television network, hasn't been on since.
