The network said in a statement that Donald Trump’s decision is “unprecedented” and accused him of “terrorizing” debate host Megyn Kelly.
Donald Trump clashed with Megyn Kelly at a Fox News debate in 2015 and had demanded she be removed from today’s panel.
The billionaire’s decision to pull out has been mocked by his Republican rivals.
The debate is the final one before the first real test of the election campaign, the Iowa caucus on February 1 when voters in the state pick their presidential nominee.
“Capitulating to politicians’ ultimatums about a debate moderator violates all journalistic standards, as do threats,” the Fox News network said in a statement.
“We’re not sure how Iowans are going to feel about him walking away from them at the last minute, but it should be clear to the American public by now that this is rooted in one thing – Megyn Kelly.”
It added: “We can’t give in to terrorizations toward any of our employees.”
Megyn Kelly accused Donald Trump of misogyny in the first debate in August 2015 and he responded the next day by accusing her of having blood “coming out of everywhere”. He denied he was referring to menstruation.
Donald Trump, who leads the Republican field, quit the debate on January 28 after Fox put out a sarcastic statement that questioned his ability to deal with Russian President Vladimir Putin.
Fox responded by accusing Donald Trump of “viciously attacking” Megyn Kelly, threatening her and spending four days trying to get her removed.
Republican presidential candidate Senator Ted Cruz mocked Donald Trump for his announcement.
Ted Cruz challenged Donald Trump to a one-on-one debate and taunted him on Twitter using the hashtag #DuckingDonald.
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