Washington (CNN) Donald Trump said Monday that Fox News anchor and debate moderator Megyn Kelly should be the one apologizing to him, and not the other way around.

"She should really be apologizing to me, to tell the truth," Trump told MSNBC's "Morning Joe."

Trump's comments mark a new layer in his continued scrapping on the "blood" comment -- which has roiled even Trump supporters and led to him being disinvited from the past weekend's conservative Red State conference.

"You could see there was blood coming out of her eyes," Trump told Lemon Friday night . "Blood coming out of her wherever."

He continued to insist Monday that anyone who assumed he was linking Kelly's debate questions to her menstrual period to be sick and said any assertion like that would be "inappropriate."

Trump swiped the spotlight at the beginning of last Thursday's debate when he was the only candidate not to rule out a third-party candidacy. News reports surfaced Monday, saying that he might reverse that position, but the Trump campaign denied them.

"There is no truth to this report and no change from what Mr. Trump has previously said," Trump spokeswoman Hope Hicks said Monday.

Trump, who has been calling in on-air to every network except Fox, tweeted Monday "Roger Ailes just called. He is a great guy & assures me that "Trump" will be treated fairly on @FoxNews. His word is always good!"

Roger Ailes just called. He is a great guy & assures me that "Trump" will be treated fairly on @FoxNews. His word is always good! — Donald J. Trump (@realDonaldTrump) August 10, 2015

Speaking on MSNBC, Trump laid out some of his reasoning for why he, himself, is not the one apologizing, pointing to continued support among Republican voters, even in the wake of the criticism.

"A lot of good things are happening, so maybe I shouldn't change anything," he said.

He said, again, that he would be "greatest" president for women if he was elected but declined to provide details. Pressed by MSNBC's Mika Brzezinski on whether he would support equal pay, Trump said he would not detail policy on the show but would have a proposal for women coming out soon.

"I don't want to discuss that on this show," he said. "I will be coming out with some policy on that."