Monday night, Trump unleashed a fresh tirade against Kelly, who he has been disparaging ever since she moderated the first GOP presidential debate earlier this month.
Fox News chairman and CEO Roger Ailes called on Trump to apologize in a statement today.
"Donald Trump's surprise and unprovoked attack on Megyn Kelly during her show last night is as unacceptable as it is disturbing. Megyn Kelly represents the very best of American journalism and all of us at Fox News Channel reject the crude and irresponsible attempts to suggest otherwise," Ailes said in his statement. "I could not be more proud of Megyn for her professionalism and class in the face of all of Mr. Trump's verbal assaults."
In his own rebuttal statement, Trump indicated he has no intention of apologizing.
“I totally disagree with the Fox statement. I do not think Megyn Kelly is a quality journalist,” Trump said. “I think her questioning of me, despite all of the polls saying I won the debate, was very unfair. Hopefully in the future I will be proven wrong and she will be able to elevate her standards to a level of professionalism that a network such as Fox deserves.”
Ahead of a planned campaign appearance in Iowa tonight, Trump and his GOP rivals also spent the day picking fights with each other.
On Twitter, Trump taunted South Carolina Sen. Lindsey Graham, who is also seeking the Republican presidential nomination, with the results of a new poll of voters in South Carolina -- an important early primary state.
"Come to South Carolina and I'll beat his brains out," Graham responded in a CNN interview today.
And on a conference call from Alaska, where he is campaigning, Kentucky Sen. Rand Paul also attacked Trump, the leader in national and many state polls, labeling his GOP opponent, a “fake conservative.”
"I think the polls are a very temporary temperature of where people are," Paul said. "And I think they’ve shown that some people are falling for someone who I believe is a fake conservative and I’ll continue to say so."
Bush and Trump also sparred on today over the issue of immigration.
During a visit to the border town of McAllen, Texas, on Monday, Bush criticized Trump's proposal to deal with the issue of illegal immigration as "unrealistic."
"You have to have a much deeper strategy than just building a fence," Bush said.
And today, Bush reinforced his critique in a statement posted on his campaign website.
"Donald Trump has staked his presidential campaign primarily on the issue of illegal immigration. But rather than offering serious solutions to this national problem, Mr. Trump is pushing a plan that is not conservative, wildly unrealistic and not reflective of our values," the former Florida governor wrote. "If that is the position put forth by our party’s standard bearer, the result will be the election of Hillary Clinton and the continuation of the disastrous economic and foreign policies of Barack Obama."