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WASHINGTON -- An unseemly feud has erupted in the Republican realm, between the party's most-influential news network and its poll-leader in the U.S. presidential primary.

It's Fox News vs. Donald Trump.

But it goes far beyond the candidate and the cable network. Their fast-escalating fight risks bloodying the entire Republican 2016 presidential campaign, if it isn't settled amicably.

Trump has repeatedly mused about mounting a third-party run -- a right-wing nightmare scenario that would splinter the conservative vote and virtually guarantee Democrats the White House.

If he's treated unfairly by the party establishment, he says, he might do it. And boy was he treated unfairly during the Republican primary debate on that network -- at least in his telling.

He's been levelling insults at network personalities since then, with those directed at female co-moderator Megyn Kelly crude enough to prompt a backlash from other Republicans.

The bombastic billionaire said the Fox crew had it in for him during Thursday's debate which, thanks largely to his presence, smashed the all-time U.S. viewership record for a cable-news program.

He responded to the perceived slight as he often does: by doubling-down on denigration.

"I'm very disappointed in Fox News," he told rival network CNN on Friday. "I think they probably had an agenda. But, certainly, I don't have a lot of respect for Megyn Kelly... She gets out, and she starts asking me all sorts of ridiculous questions. And you know, you can see there was blood coming out of her eyes, blood coming out of her wherever."

It was that last phrase -- about her "wherever" -- that had some conservatives saying he'd finally taken his mean, often-misogynistic shtick too far.

Trump was booted out of a planned grassroots event Saturday. A rival presidential candidate, former Hewlett-Packard CEO Carly Fiorina, tweeted: "Mr. Trump: There. Is. No. Excuse," and, "I stand with Γåòmegynkelly." He responded with a press release that called the event organizer a "total loser" who'd said far more sexist things.

The debate brought to the surface a tension that's bubbled within the party, amid Trump's meteoric rise to the top of the Republican primary polls.

That tension was hinted at weeks ago when Fox owner Rupert Murdoch suggested Trump was embarrassing his party. However, the media mogul had tamped down his criticism as the Republican establishment wrestled with how to defuse the human grenade in their midst.

So most of the field, and the conservative commentariat, has been handling him with care -- cringing silently through every outrage to avoid poking the man a rival campaign likened to a rattlesnake.

But that appeared to change in Thursday's debate, which attracted a record audience of 24 million.

Moderators opened the debate by quizzing Trump on a third-party run. Kelly read a list of past insults he'd levelled against women including "fat pig" and "dog." Other candidates faced tough questions but none were pressed as hard as Trump.

Once the debate was over, the network ran a focus group where participants trashed Trump's performance.

The billionaire's backers sprang to his defence, slamming the focus-group leader Frank Luntz, and unleashing a torrent of insults on Kelly's Facebook page.

Trump himself led the counter-attack. He called Luntz a slob, and tweeted at Fox News: "You should be ashamed of yourself. I got you the highest debate ratings in your history & you say nothing but bad."

Liberals leaned back and enjoyed the show.

One writer called the debate the funniest political program in American history. Matt Taibbi also called it a well-deserved reckoning for the Grand Old Party, which has fed a fear of foreigners that Trump is now exploiting.

"The Republican party and its allies at Fox, on afternoon radio and in the blogosphere have spent many years now whipping audiences into zombie-style bloodlusts," wrote the Rolling Stone writer.

"When it suited them, party insiders told voters across middle America that foreigners were trying to crawl through their windows to take their wives, and that stuffed suits in Washington and in the media were conspiring to enslave their children in Marxist bondage.

"Now all of that paranoia is backing up on them. They created this monster, and it's coming for them now. Trumpenstein lives. He is loose in the town and on his way to the doctor's castle. We may not be laughing two years from now, but for the time being, man, what a show."