"You're fired", Mr Cruz said, riffing on Mr Trump's famous rejoinder from his reality TV show The Apprentice. He has made a decision to boycott Thursday night's debate after Fox News refused to remove its anchor, Megyn Kelly - with whom Trump famously clashed at the first Republican debate in August - from moderating Thursday night's debate.

The New York real estate mogul's presence has helped produce massive ratings in the previous six Republican presidential debates.

Cruz, Trump's main rival in Iowa, used the hashtag #DuckingDonald to make fun of Trump for ducking out of the debate and tweeted a mocked-up picture of Trump's head on Donald Duck's body sitting on a pile of money. Trump has pledged to hold a separate event Thursday night while the other candidates debate.

Mr Trump's withdrawal from the debate was seen as a bold gamble. During his appearance on Bill O'Reilly's show, he showed no signs of relenting saying, "I don't like being taken advantage of, in this case I was being taken advantage of by Fox".

"I think I could make the argument - we'll see how things progress with the campaigns across the country - that this could be the biggest debate of the entire year".

"I have zero respect for Megyn Kelly", Trump said.

O'Reilly then tried appealing to Trump's patriotism, but also failed on that front.

Trump campaign manager Corey Lewandowski said in a pair of television interviews Wednesday that Trump, a billionaire businessman, "knows when to walk away from a bad deal".

Donald Trump appeared Wednesday night on Fox News' The O'Reilly Factor to air his grievances with the network.

Cruz also said he has a reservation at Western Iowa Tech Community College for Saturday at 8 P.M. for Trump to debate him one-on-one in Sioux City, Iowa.

Trump said he subscribed to a different tenet. "He's won every debate", Lewandowski said.

While Webber doesn't support Trump, he said he was disappointed that the Iowa frontrunner planned to sit out the final Republican debate ahead of the February 1 caucuses.

"Frankly, you know, in the past, anytime somebody thought he did something that cost him, it didn't cost him", Kass said.

Florida Sen. Marco Rubio, meanwhile, said he didn't "have time for these kinds of distractions".

The feud between Trump and Fox reignited this week after the Republican front-runner began balking at the prospect of taking part in Thursday's debate, claiming once again that O'Reilly's colleague Megyn Kelly was treating him unfairly.