Trump contended on CBS' "Face the Nation" that it's former Florida governor Job Bush who has the problem with women, thanks to a comment Bush made that last week when discussing cutting off federal money for Planned Parenthood.
"I'm not sure we need half a billion dollars for women's health issues," Bush said. He later issued a statement saying he had misspoken and was referring only to the "hard-to-fathom $500 million in federal funding" for Planned Parenthood.
"I think he's got a huge problem," Trump said. He argued that it was worse than a video recording of 2012 GOP nominee Mitt Romney in which Romney said that 47% of voters were dependent on the government and would vote for President Barack Obama, no matter what.
Trump said he would be "very much up" on issue of women's health" and "very much into the whole thing of helping people and helping women."
The billionaire real-estate magnate's assertions about Bush come after Trump was widely criticized for comments he made about Fox anchor Megyn Kelly, who moderated Thursday night's GOP debate. Kelly posed tough comments for Trump on Thursday and pointed out that he'd called women "fat pigs, dogs, slobs, and disgusting animals."
The following night, Trump went after Kelly in a CNN interview where he seemed to suggest she was being tough on him because she had her period.
"You could see there was blood coming out of her eyes. Blood coming out of her wherever," Trump said.
The only woman in the GOP race said she believes women are "horrified" by Trump's offensive and that Trump may be unprepared for the pressure that comes with being president.
Trump's unconventional, insurgent campaign has excited many anti-establishment conservatives while confounding party leaders already facing the prospects of a bruising fight among 17 candidates.
The latest controversy started Thursday night when Fox News debate moderator Megyn Kelly recounted Trump's history of incendiary comments toward women.
Angry over what he considered unfair treatment at the debate, Trump told CNN on Friday night that Kelly had "blood coming out of her eyes, blood coming out of her wherever."
That remark cost Trump a prime-time speaking slot at the RedState Gathering, the Atlanta conference where several other presidential candidates spoke to about 1,000 conservative activists.
RedState host Erick Erickson said in a statement that Trump had violated basic standards of decency, even if his bluntness "resonates with a lot of people." The Trump campaign retorted by calling Erickson a "total loser" who backs other "establishment losers."
In the first of four interviews on the Sunday news shows, Trump stuck to his assertion that only "a deviant" would interpret his comment beyond a harmless barb, and dug in against Kelly and Erickson.
Jeb Bush, the presidential favorite for many top Republican donors, said at RedState that Trump's bombast would hurt the GOP's chances with women, who already tilt toward Democrats in presidential elections. "Do we want to win? Do we want to insult 53 percent of our voters?" the former Florida governor asked.
Other candidates criticized Trump; some sought to avoid giving him more of their time.
Former Hewlett-Packard CEO Carly Fiorina, the only woman running for the GOP nomination, was among the first to speak out.
"I think women of all kinds are really sort of horrified by this," she told CBS, saying there is a difference between being politically incorrect and "hurling personal insults at all kinds of people."
She also suggested that Trump's reaction was a sign that he is too thin-skinned. "If you think a question is tough, imagine the pressure of actually being in the Oval Office," she said.
Florida Sen. Marco Rubio said he will let Trump answer for his own words.
"If I comment on everything he says, my whole campaign will be consumed by it," Rubio told NBC's "Meet the Press."
"He says something every day," Rubio said.
Mike Huckabee, a former Arkansas governor, seemed exasperated by it all, at one point snapping at reporters after being asked several Trump-related questions. "I'm running for president," he said. "I'm not running for social media critic of somebody else who's running for president."
By Saturday evening, Trump's campaign announced that he had fired a top consultant, Roger Stone. Stone said on Twitter that he had "fired Trump," not the other way around.
According to an email obtained by the Associated Press, Stone wrote to Trump, "The current controversies involving personalities and provocative media fights have reached such a high volume that it has distracted attention from your platform and overwhelmed your core message."
Trump's campaign manager said he never received that message.
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Barrow reported from Atlanta.