For months, Donald J. Trump had kept up a relentless and sometimes personal barrage against the Fox News anchor Megyn Kelly, in retaliation for the tough — and, he said, unfair — questioning she submitted him to in the first Republican debate last summer. And for months, Fox News and Ms. Kelly had avoided taking the bait, intent on sticking to Ms. Kelly’s vow to avoid becoming part of the story.

Fox‘s decision to fight back this week after Mr. Trump demanded Fox remove her from its panel of moderators at a debate scheduled for Thursday sent the already unconventional Republican race for president into uncharted territory just days before the first critical contest in Iowa.

With Mr. Trump declaring he will skip the debate, the most potentially consequential fight on the campaign trail is not between Mr. Trump and his fellow candidates, but between Mr. Trump and the top-rated cable television news network, which is also one of the most important forces in Republican politics.

Neither Fox News nor Mr. Trump showed any indication of budging on Wednesday. That was in spite of furious discussions inside the network, inside the campaign, and in direct and indirect communications between the two sides. The talks even included a conversation between Mr. Trump‘s daughter, Ivanka, and the influential Fox News chairman, Roger Ailes, who contacted her to gauge Mr. Trump‘s seriousness but, network officials said, did not seek to change his mind.

Mr. Trump said the final straw was a sarcastic press statement Fox News released on Tuesday, which jokingly questioned how he would handle President Vladimir V. Putin of Russia and Ayatollah Ali Khamenei of Iran if he could not handle Ms. Kelly. On Twitter on Wednesday evening , Mr. Trump called it «childishly written» and «taunting.» But network officials said the statement was, in fact, an attempt at levity composed by Mr. Ailes, who they said had reached his limit with Mr. Trump‘s rants against Ms. Kelly, which ramped up over the weekend. Still, Fox‘s stance was primarily about journalistic principles, the officials said.

Fox News said it would never accede to Mr. Trump‘s demand that it remove Ms. Kelly from its panel of moderators, which also includes the network anchors Bret Baier and Chris Wallace. Though the move could cost Fox News debate-night ratings, officials there said Rupert Murdoch, the executive co-chairman of the network’s parent company, 21st Century Fox, lent Mr. Ailes his support when the two spoke by phone, after the episode exploded into a public spectacle. (Mr. Murdoch made his own political news on Wednesday, posting on Twitter his support for an independent presidential run by former Mayor Michael R. Bloomberg of New York.)

On Fox News’s «The O’Reilly Factor» on Wednesday night, Mr. Trump once again went after Ms. Kelly, saying that he has «zero respect» for her. But he also placed blame on the network, saying, «This also had to do with the stupidity of Fox and what they did.»

«I was not treated well by Fox,» said Mr. Trump, who has made scores of appearances on Fox News and Fox Business. «They came out with this ridiculous P.R. statement — it was like drawn up by a child. And it was a taunt. And I said, ‘How much of this do you take?’ »

The interview with the anchor Bill O’Reilly made for provocative television, featuring some heated exchanges and moments when the two men simply talked over each other. Mr. Trump chided Mr. O’Reilly for breaking what he described as a preinterview agreement the two had about not asking whether he would change his mind and join the debate. «I told you upfront: Don’t ask me that question,» Mr. Trump said.

Mr. O’Reilly said that was the truth, and complimented him for coming on.

In shunning the debate, Mr. Trump is making a gamble that Fox News’s grip on the voters to whom he appeals — populists who are fed up with prescribed rules of engagement — is not strong enough to hurt him. And while Mr. Trump received support from his legion of Twitter followers and some talk radio hosts, including Rush Limbaugh, his rivals openly wondered whether he would reverse course. Jeb Bush told reporters in Iowa he had placed a $20 bet that Mr. Trump would appear after all.

Aides to Mr. Trump, however, said they were moving forward with plans for a competing event to benefit veterans on Thursday.

The dispute between Fox and Mr. Trump perfectly encapsulated the conundrum Mr. Trump has presented to the news media in the prevoting phase of the campaign season. He is a presidential front-runner who consistently, singularly, delivers hit-level ratings to whatever network he appears on. Those ratings, of course, translate into revenue, giving Mr. Trump that sine qua non of all winning negotiators: leverage. The result over several months has been push-pull between the networks’ dual responsibilities to their bottom lines and their own journalistic standards, from which Mr. Trump frequently challenges them to stray.

Late last year, Mr. Trump suggested he might demand that CNN pay him $5 million to appear on its debate (he said the money would go to veterans, but did not press the demand). Earlier this month, Mr. Trump took credit for persuading ABC News to cut The Union Leader of Manchester out of the debate they had planned to sponsor together, citing the newspaper’s harsh editorials against him (ABC said that The Union Leader’s tensions with Mr. Trump had become a distraction).

«As a result of a number of different forces, the media have in fact lost control and in losing control, they’ve also lost some of the dignity that used to attend a presidential process,» said Frank Sesno, a former CNN Washington bureau chief and now the director of the School of Media and Public Affairs at George Washington University. «It’s an anti-institutional election cycle, and that’s the reason that Trump and all the others have gained so much traction — one of the institutions that they’re running against is the media.»

In his latest debate standoff, Mr. Trump was making Fox News an offer it had to refuse: his presence at its debate in return for veto power over Ms. Kelly. Making matters worse, Bill Sammon, the network’s Washington managing editor, said Mr. Trump‘s campaign manager, Corey Lewandowski, had appeared to level a threat against Ms. Kelly in a conversation they had over the weekend. In arguing against Ms. Kelly, Mr. Sammon said, Mr. Lewandowski had mentioned that Ms. Kelly had a «couple of rough days» after the first debate last summer and he would hate to see a repeat, prompting Mr. Sammon to warn against such «veiled threats.» Mr. Lewandowski was referring to the vitriol against Ms. Kelly by Mr. Trump‘s supporters, not to mention the candidate himself, who said that during the debate, «there was blood coming out of her eyes, blood coming out of her wherever» — a remark widely interpreted as a reference to menstruation.

Mr. Lewandowski did not respond to questions about Fox complaints about him, but told MSNBC on Wednesday that it was a private conference call with the network and «you’d hope that they’d keep that conversation private.»

Mr. Ailes decided to move forward with the statement involving Mr. Putin and the ayatollah after Mr. Trump posted a video on an Instagram account calling Ms. Kelly «really biased against me,» and asking his followers, «Do you really think she can be fair at a debate?» The subsequent Fox News statement read, «We learned from a secret back channel that the Ayatollah and Putin both intend to treat Donald Trump unfairly when they meet with him if he becomes president — a nefarious source tells us that Trump has his own secret plan to replace the Cabinet with his Twitter followers to see if he should even go to those meetings.»

Mr. Trump was deeply disappointed and bothered by the personal nature of the statement, according to two people who spoke to him, in part because in his view, it was the type of statement a political campaign, not a television news organization, would make. Still Mr. Trump did not hesitate to get highly personal with Ms. Kelly again on Wednesday, saying on Twitter, «I refuse to call Megyn Kelly a bimbo, because that would not be politically correct. Instead I will only call her a lightweight reporter!»

The political world was split on the likely effect of the choice to sit out the debate — with rival campaigns saying it would hurt him by giving their candidates the stage to themselves and others, like Mr. Limbaugh, saying it might help him. Some television pundits wondered whether Mr. Trump was trying to rob his chief rival in Iowa, Senator Ted Cruz of Texas, of a chance to strike a dynamic-shifting blow against Mr. Trump.

Mr. Cruz’s campaign leapt at the opportunity to portray it as a slight to Iowans. At a rally in Iowa on Wednesday night, Mr. Cruz said, «What does it say when Donald tells the men and women of Iowa, ‘My time is more important than your time?’ »

According to several people who have spoken with him, Mr. Trump does not believe he will lose Iowa.

With No Détente, Donald Trump’s Feud With Fox Takes Center Stage