Megyn Kelly has no intention of apologizing to Donald Trump, even though the presidential hopeful demanded a mea culpa from the Fox News host, who had asked him about calling women "fat pigs, dogs, slobs, and disgusting animals" during last week's first Republican debate.

"Mr. Trump was upset with a question I asked him at the debate last week about his electability, and specifically comments he has made in the past about women," Kelly said, via Entertainment Weekly. "Apparently, Mr. Trump thought the question I asked was unfair, and felt I was attacking him. I felt he was asked a tough but fair question. We agree to disagree."

The back-and-forth between Trump and Kelly has filled newspaper columns this week after the real-estate mogul called the moderator's inquiry "ridiculous" and alleged that "there was blood coming out of her eyes ... blood coming out of her wherever." Some observers interpreted his comments as a veiled reference to menstruation, an allegation Trump has denied.

"Who would say that?" Trump told CNN's Jake Tapper on "State of the Union" on Sunday. "Do you think I'd make a statement like that? Who would make a statement like that? Only a sick person would even think about that."

The "Apprentice" star, who has been married three times, insisted that he "cherished" women and went on to dub Erick Erickson a "loser." The conservative blogger had uninvited Trump from the weekend's RedState Gathering over his latest controversial comments, the news channel recalled.

Texas Sen. Ted Cruz, one of Trump's challengers in the race for the 2016 GOP nomination, meanwhile, warned his fellow Republicans that slamming the front-runner may be a dangerous game, Politico reported.

"Donald Trump had a rally in Phoenix, Ariz., (at which) between 10,000 and 20,000 people came out," Cruz explained. "When you attack and vilify the people at that rally as crazies, it does nothing to help Republicans win in 2016. I'd like every single person at that rally to show up and vote in 2016, knock on doors with energy and passion, and turn this country around."