(CNN) Tweet by tweet, Donald Trump could undermine the GOP's efforts to close the gender gap, a strategy that is critical to the party's hopes of retaking the White House in November.

Miffed that an anti-Trump super PAC targeted his wife by using a racy photo of her, Trump has gone after Ted Cruz's wife, upending the conventional rules of what's acceptable in presidential politics.

He posted on Twitter Tuesday that he would "spill the beans" on Heidi Cruz, a former Goldman Sachs executive.

Lyin' Ted Cruz just used a picture of Melania from a G.Q. shoot in his ad. Be careful, Lyin' Ted, or I will spill the beans on your wife!

And he re-tweeted an unflattering photo late Wednesday of Heidi Cruz next to his wife that was captioned "a picture is worth a thousand words."

An angry and emotional Ted Cruz ripped into Trump Thursday for the attacks and repeatedly declined to say whether he would support Trump if he's the Republican presidential nominee.

"Donald, you're a sniveling coward and leave Heidi the hell alone," he told reporters in Wisconsin, calling on Republicans to unite around his candidacy and defeat Trump.

He added that "strong women scare Donald."

Why I'm voting for Donald Trump



Trump's tweets about Heidi Cruz are the latest -- and perhaps most provocative -- salvo in a string of messy dust-ups between Trump and prominent women that could complicate the GOP's general election chances. With Hillary Clinton as the likely Democratic nominee, Trump's history of insults aimed at women could be even more problematic, injecting gender into the race in unpredictable ways. Trump's tweets about Heidi Cruz are the latest -- and perhaps most provocative -- salvo in a string of messy dust-ups between Trump and prominent women that could complicate the GOP's general election chances. With Hillary Clinton as the likely Democratic nominee, Trump's history of insults aimed at women could be even more problematic, injecting gender into the race in unpredictable ways.

"He has a 30-year pattern of this kind of sexism against women and lashing out against women when he doesn't get his way," GOP strategist Tara Setmayer said in an interview Wednesday with Don Lemon on "CNN Tonight." "So, you're going to have a tough time defending this if, God forbid, he's our nominee in the general (election). Do you think Hillary Clinton is going to let this kind of stuff go? This is mild in comparison to what's going to happen to him in the general."

The Trump campaign did not respond to an email seeking comment on his tweets about Heidi Cruz.

Critics could have plenty of material if they aim to portray Trump as the newest leader in the so-called "war on women."

Trump famously fought with Fox News host Megyn Kelly, an entanglement that her network has called a "sick obsession."

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He once called Rosie O'Donnell a "fat pig." And he sent a note to New York Times columnist Gail Collins that said she had the "face of a dog."

He also criticized former Republican presidential rival Carly Fiorina over her looks, a fight that led Fiorina to say in a debate, "Women all over this country heard very clearly what Mr. Trump said."

JUST WATCHED Fiorina: It's still different for women Replay More Videos ... MUST WATCH Fiorina: It's still different for women 01:11

The anti-Trump group Our Principles PAC recently released a video of women reading statements Trump made about women that centered on their looks.

"Bimbo. Dog. Fat Pig," the women say in the ad, adding a string of comments Trump has made over the years. "This is how Donald Trump talks about our mothers, our sister, our daughters."

Asked Monday by CNN's Wolf Blitzer about his statements about women, Trump said some of his comments were just "show business," and that he would be better for women than Clinton.

"Nobody respects women more than I do," he said. "Nobody takes care of the women -- and they take care of me because they do such a great job."

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In 2012, President Barack Obama won women overall by 11 points, though he lost married women by a similar margin.

As a result, the Republican National Committee vowed to do better, suggesting message training for candidates and a focus on "developing a forward-leaning vision for voting Republican that appeals to women" in a post-2012 report.

Yet, with his stream of insults directed at women, Trump is effectively burying that autopsy. A CNN/ORC poll released Monday shows Clinton leading Trump 60% to 33% among women in a hypothetical general election match-up.

Trump's supporters have argued that Trump is an equal opportunity insulter, attacking men and women alike over their looks. For instance, he has criticized Sens. Rand Paul and Marco Rubio over their height. He frequently lampooned Rubio's penchant for sweating under the lights of the debate stage.

But even some of his biggest supporters have criticized him over his latest comments.

"I do wish he would have stayed above the fray and not pulled Heidi into this because the way I see it is Melania Trump is a very respectable woman, speaks five languages, very well educated," CNN contributor and Trump supporter Kayleigh McEnany told Don Lemon. "Heidi Cruz, likewise, very smart lady. Both of these women are model women, I would say."

His remarks about women have not hurt him in the nomination fight.

Among Republicans, Trump leads with women, getting 44% to Ted Cruz's 32%, according to a recent CNN/ORC poll. In the near term, closing that gap could be one way for Cruz to boost his chances of slowing Trump's march to the 1,237 delegates he needs to win the party nomination.

A Thursday morning exchange with Ben Carson on "The View," a daytime show that draws hundreds of thousands of female viewers, offered a preview of the kinds of questions Republicans will face if Trump is their nominee.

"You have allied yourself with a man who has bashed women, made countless racist remarks, and you're Ben Carson, why would you align yourself with that?" Whoopi Goldberg asked.

"Well, you know, you have to look at the good and the bad. There is no perfect person," Carson said, adding that Trump helped integrate clubs in Palm Beach. "Has he said some things that I wouldn't say or that you wouldn't say? Of course."