Donald Trump's campaign for president has suddenly become a campaign against Fox News.

One day after he starred in Fox's GOP primary debate, Trump lashed out at debate moderator Megyn Kelly and the network as a whole.

In an interview with Don Lemon on "CNN Tonight," Trump on Friday accused Fox of asking "vicious," unfair questions at the debate and called Kelly "overrated."

Kelly had pressed Trump about misogynistic, sexist comments in his past.

On CNN, Trump called the questions "ridiculous," called her "off-base," and said, "You could see there was blood coming out of her eyes. Blood coming out of her wherever."

Trump told Lemon that he didn't know whether he'd participate in a future Fox debate. "I might not, to be honest," he said. "I didn't think they were fair."

Kelly and Fox News declined to comment on Trump on Friday night. Privately, people at the network are disgusted, although they don't want to be seen as quarreling with Trump.

Following Trump's CNN interview, Fox owner Rupert Murdoch tweeted: "Baier, Kelly, Wallace great job Thursday. Fine journalism, no more, no less. Friend Donald has to learn this is public life."

On Saturday Trump's campaign said he meant Kelly's "nose" when he said "wherever," and "only a deviant would think anything else."

Before his interview with Kelly, Trump had blasted Kelly on Twitter and wrote that Fox News "should be ashamed."

He wrote: "I got you the highest debate ratings in your history & you say nothing but bad..."

His also took aim at Fox News analyst Charles Krauthammer, who said the Thursday night debate revealed Trump to be "testy" and "thin-skinned." Trump said Fox should fire him.

The hatred that clown @krauthammer has for me is unbelievable -- causes him to lie when many others say Trump easily won debate. — Donald J. Trump (@realDonaldTrump) August 7, 2015

It almost goes without saying that Fox's tough questioning of Trump made for good TV, which showed in the ratings. A total of 24 million viewers tuned in, making it by far the highest-rated primary debate in TV history.

But many of Trump's supporters are now lashing out against Fox -- some are even calling for a boycott -- and accusing the network of biased coverage.

More broadly, the television news world is buzzing about whether Fox News has turned against Donald Trump.

Is Fox News chairman Roger Ailes strategically positioning his network against the candidate, either to influence the Republican party or simply spark his ratings? Or is the network just covering the news the way it's supposed to?

Trump's anti-Fox tweet storm: After sparring with the Fox moderators during Thursday's debate, Trump stayed up late and attacked them on Twitter, directing most of his ire at Kelly.

"The Fox News trio, especially Megyn Kelly, was not very good or professional!" he wrote at 3:53 a.m. Friday.

Trump also said she "really bombed," and he retweeted a dozen fans who criticized her hostility and called her a "bimbo."

Then he called into MSNBC's "Morning Joe" to advance his complaints.

"I'm very surprised at Fox News," he said, accusing the moderators of being tough on him while going easy on others. "I would say it's pretty unprofessional, but we will live with it."

The same questions that ticked off Trump impressed a lot of journalists, even some who rarely say anything nice about Fox News.

"Hooray for Fox News," New York Times columnist Frank Bruni wrote, while admitting he "never imagined writing" those words.

Related: Fox moderators get applause for tough questions

"Hidden agenda" speculation: In recent weeks there has been a lot of curiosity about Fox's Trump coverage, given the channel's importance in conservative circles and Murdoch's criticism of Trump.

In mid-July Murdoch called the GOP frontrunner "embarrassing." Then New York magazine reported that Murdoch had asked Ailes to dial back the channel's Trump coverage and Ailes had declined.

Ailes, however, threw cold water on that report. And Murdoch has more recently softened his tone about Trump.

Nevertheless, the notion of Fox being at "war" with Trump has taken hold. Some have speculated that Ailes wants to knock out Trump to clear the way for a more electable Republican.

One of the viewers Trump retweeted overnight wondered what Kelly's "hidden agenda" might have been.

On Thursday night Kelly's first question to Trump was as follows:

"You've called women you don't like 'fat pigs, dogs, slobs, and disgusting animals.'" He interrupted to say "only Rosie O'Donnell," and she shut him down: "No, it wasn't."

She continued: "Your Twitter account has several disparaging comments about women's looks. You once told a contestant on 'Celebrity Apprentice' it would be a pretty picture to see her on her knees. Does that sound to you like the temperament of a man we should elect as president, and how will you answer the charge from Hillary Clinton, who was likely to be the Democratic nominee, that you are part of the war on women?"

He didn't really answer her question.

Anti-Trump focus group: Trump's battles with Kelly will probably further burnish the anchorwoman's reputation as a tough-as-nails television star. The conservative cable news channel has sought to market Kelly as a strong and independent voice.

Inside Fox, the view at the end of the debate was that all three moderators -- Kelly, Bret Baier and Chris Wallace -- had fared well, and that Kelly was the stand-out.

Some political observers said that Fox's post-debate coverage seemed tilted against Trump, partly because of a Frank Luntz focus group, convened by Fox and televised immediately after the debate, that had an anti-Trump tone.

Trump was incensed by the focus group.

"Where did you find that dumb panel" he asked Luntz via Twitter.

He also tweeted that Luntz is "a low class slob who came to my office looking for consulting work and I had zero interest. Now he picks anti-Trump panels!"