Paulina disappeared from the segment, which ended early, Levanthal noted. He apologised to viewers over her language.
The incident prompted an additional on-air apology from the network.
"We are all Americans, and that is the focus, and we want to reiterate that we do not condone the language that Anna Paulina just displayed here, and we apologise to Secretary Clinton for that," Arthel Neville said. "Fox News does not condone her sentiment."
A spokesperson for Clinton did not return comment.
It's not clear if Paulina, who has appeared as a guest on Fox News in the past, will be invited back. A spokeswoman for the network pointed to Neville's apology and declined to comment further.
Paulina is the director of Spanish engagement at Turning Point USA, a conservative group first built to fight a perception of liberal bias at college campuses. It has since become a safe space for young conservatives.
The group did not return a request for comment, and its website features only one mention of Paulina from an event she spoke at in October.