"I certainly will not apologize for doing good journalism, so I'll continue doing my job without fear or favor," she said at the start of her Fox News show, The Kelly File.
While not addressing Trump's statements specifically, she added, "This is a tough business and it's time now to move forward."
Kelly had questioned the GOP candidate during Thursday's debate on his history of derogatory remarks about females – calling them "fat pigs," "slobs" and "disgusting animals."
The following day the businessman told CNN that while Kelly was questioning him, "You could see there was blood coming out of her eyes. Blood coming out of her wherever."
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He was subsequently booted from a gathering of major conservatives in Atlanta on Saturday, where he had been invited to speak, and later defended his controversial "blood" comment, insisting that "only a deviant would say" that he was referring to Kelly menstruating and being hormonal.
Others in the GOP field, including Carly Fiorina, condemned his remarks, and on Monday Democratic presidential frontrunner Hillary Clinton described his comments as "offensive."