The question for Fox News comes down to whether to hold Mr. Trump accountable for the things he has said and done, or whether its viewers would prefer to bask in “Trumpism,” at least while it’s ascendant.

Complicating the matter further for Fox News is a history of enmity between Mr. Trump and Rupert Murdoch, the founder of 21st Century Fox, which owns the network. Most recently, Mr. Murdoch criticized the candidate for mocking Senator John McCain, Republican of Arizona, for being captured by the enemy during the Vietnam War. “When is Donald Trump going to stop embarrassing his friends, let alone the whole country?” Mr. Murdoch wrote on Twitter.

Fox News has been tough on Republican presidential candidates in the past. At a debate in 2011, the network’s Bret Baier put the entire G.O.P. field on the spot, asking the candidates if they would accept a debt-reduction deal in which they would receive $10 in spending cuts for every $1 in tax increases. (None of the eight candidates said they would.)

During the same campaign, Mr. Baier grilled the Republican front-runner, Mitt Romney, on his Fox show about changing his stance on a wide range of issues, from climate change to gay rights. Mr. Romney later described the interview as “overly aggressive” and “uncalled for.”

Image Megyn Kelly at the Republican debate. Credit... John Minchillo/Associated Press

But the battle between Mr. Trump and Ms. Kelly has been something else altogether. To begin with, Ms. Kelly was pressing Mr. Trump not on his political positions but on his character, and more specifically his treatment of women.

“You’ve called women you don’t like fat, pigs, dogs, slobs and disgusting animals,” she said during the debate. It seems safe to say that Mr. Trump’s response, in an interview the following night on CNN — that she had “blood coming out of her wherever” when questioning him — brought us into uncharted political waters. (Mr. Trump says his comments were not a reference to menstruation.)