The businessman denied implying that Megyn Kelly, a moderator at last Thursday's presidential debate, was tough on him because she was menstruating.

He has been castigated since telling CNN on Friday: "You can see there was blood coming out of her eyes, blood coming out of her wherever."

Mr Trump, who was barred from an important gathering of conservative activists this weekend in Georgia because of the jibe, was back on CNN on Sunday defending himself.

Only a "deviant" would have thought he was talking about Ms Kelly's period, he said, insisting he was talking about her nose.

"I cherish women," Mr Trump said in a phone interview with the cable network's State of the Union news programme. "I want to help women."

"Who would make a statement like that? Only a sick person would even think about it."

Mr Trump also doubled down on his criticism of Ms Kelly, refusing to apologise and saying she had asked "unfair" questions of him during the debate.

Also on Sunday, he said on ABC's This Week: "I've had such an amazing relationship with women in business.

"They are amazing executives. They are killers. They are phenomenal."

Mr Trump's top political adviser Roger Stone left the campaign on Saturday.

He said on Twitter he disagreed with the "diversion to food fight with @megynkelly".

Mr Stone said he quit after Mr Trump ignored his advice to focus on policy issues, not personal attacks. Mr Trump said he fired Mr Stone.

But the female co-chair of Mr Trump's political campaign in the election battleground state of Iowa leapt to his defence.

Tana Goertz said the businessman's attacks on Ms Kelly were "fair game".

"If you mess with the bull you're going to get the horns, sweetheart," she said.

Several of Mr Trump's rivals for the Republican nomination have seized the opportunity to score points against him.

"There's no excuse for this," former Hewlett-Packard chief executive Carly Fiorina - the only woman in the Republican field - told Fox News Sunday.

"You don't get things done by insulting people."

On Saturday, Jeb Bush told the Red State gathering which disinvited Mr Trump: "Do we want to win? Do we want to insult 53% of all voters?

"What Donald Trump said was wrong."

Mr Trump hit back by pointing out that Mr Bush risked alienating female voters last week when he said women's health didn't need $500m of federal funding.

The Republican establishment has watched Mr Trump's rise to the top of the presidential race with alarm.

Party chiefs fear his remarks about women and immigrants could damage the GOP brand and turn off many voters.

They are hoping his campaign will ultimately collapse under the weight of his incendiary rhetoric.