Fox News Channel anchor Megyn Kelly said on Monday night that she would not apologize to Republican presidential contender Donald TrumpDonald John TrumpNFL freezes policy barring players from protesting during anthem McConnell spokesman on Putin visit: 'There is no invitation from Congress' Petition urges University of Virginia not to hire Marc Short MORE for questioning him at last week's presidential debate on past comments he had made concerning women.

"I certainly will not apologize for doing good journalism," she said on her program.

Trump ignited a media firestorm over the weekend while insisting he had been treated unfairly by Kelly at last week's first Republican presidential debate, at which she was a moderator. Trump called in to CNN and said of Kelly, “You could see there was blood coming out of her eyes, blood coming out of her wherever.”

Many of the Republican presidential candidates have blasted Trump for his remarks, and conservative activist Erick Erickson barred him from participating in a cattle call of GOP contenders in Georgia over the weekend.

Trump has refused to apologize.

Kelly took to the air on Monday night for the first time since Trump made the blood remarks.

“Mr. Trump did interviews over the weekend that attacked me personally,” Kelly said at the top of her show. “I have decided not to respond.”

On Monday she did respond, however, to Trump's allegations that he was mistreated and asked unfair questions at the debate.

Trump has been lambasting Kelly and the network for dredging up past comments he had made about women.

At the debate, Kelly asked Trump if he had the temperament to be president, given comments he's made about women.

“You’ve called women you don’t like fat pigs, dogs, slobs and disgusting animals,” Kelly said to Trump at the debate.

Kelly on Monday night defended the line of questioning.

“Apparently Mr. Trump thought the question I asked was unfair and that I was attacking him,” Kelly said. “I felt he was asked a tough but fair question. We agree to disagree.

“Mr. Trump is an interesting man who has captured the attention of the electorate,” Kelly continued. “That’s why he’s leading in the polls. Trump, who is the front-runner, will not apologize, and I certainly will not apologize for doing good journalism. So I will continue doing my job without fear or favor, and Mr. Trump, I expect, will continue with what has been a successful campaign so far. This is a tough business and it’s time now to move forward.”

Trump did not respond to requests for further comment.

It appears that the escalating rhetorical war between Fox News and Trump is now cooling off.

Fox News CEO and Chairman Roger Ailes said in a memo obtained by CNN on Monday that he had a “blunt but cordial” conversation with Trump.

"Donald Trump and I spoke today," Ailes said in the statement. "We discussed our concerns, and I again expressed my confidence in Megyn Kelly. She is a brilliant journalist and I support her 100 percent.

“I assured him that we will continue to cover this campaign with fairness & balance,” Ailes continued. “We had a blunt but cordial conversation and the air has been cleared."

Trump will return to Fox News for the first time since the debate on Tuesday morning’s edition of "Fox & Friends."

The billionaire real estate mogul is leading all GOP contenders in the early stages of the race for the Republican nomination.