JERSEY CITY, N.J. (CBSNewYork/AP) -- Donald Trump returned to Fox News Tuesday morning for the first time since attacking the network's Megyn Kelly in the wake of her tough questioning of the GOP candidate for president during the party's first debate.
But he made no mention of her or the debate during the appearance.
In a phone interview with "Fox & Friends'' host Steve Doocy, Trump made only passing reference to his brouhaha with the network, saying that they'd "always been friends.''
He also refused once again to rule out a third-party run.
"We're going to keep the door open, we're going to see what happens,'' he said. "I want to run as a Republican but I do want to keep that door open in case I don't get treated fairly.''
Donald Trump Returns To Fox News After Flap
Trump and Fox News, one of the most powerful voices in Republican and conservative circles, have been in a bitter feud since last week's presidential debate, when Trump was questioned about his history of insults aimed at women.
While he leads recent polls, the former reality television star has drawn heated criticism from many in his own party for saying Kelly had "blood coming out of her eyes, blood coming out of her wherever.''
Trump told CNN's Chris Cuomo Tuesday morning that he doesn't owe Kelly or women in general an apology. He also said he'd be a better president for women than anyone else, 1010 WINS' Steve Kastenbaum reported.
"I cherish women, I will be so good to women, I will work hard to protect women, and I'll tell you what, work hard to protect everybody," Trump said, adding that he'd consider a law requiring equal pay for women.
The feud seemed to show signs of improvement Monday when the network's chairman, Roger Ailes, reached out to Trump directly, assuring him he'll be "treated fairly'' by the network, Trump tweeted Monday.
In a statement Monday evening, Ailes described the conversation as "blunt but cordial'' and said the air had been cleared.
Addressing what she referred to as the "dustup'' for the first time on her show Monday night, Kelly defended what she described as a "tough but fair question'' during the debate.
"We agree to disagree,'' she said.
"Trump, who is the front-runner, will not apologize. And I certainly will not apologize for doing good journalism,'' Kelly said. "So I'll continue doing my job without fear or favor. And Mr. Trump, I expect, will continue with what has been a successful campaign thus far.''
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