Throughout the night, Fox News deployed slides and videos to fact-check the candidates’s statements, and to show what appeared to be a reversal of positions on various issues. And during the occasionally raucous debate, Ms. Kelly said “stand by” seven times to interrupt cross-talking candidates, something that many people on social media said was a remarkably polite way to say “shut up” (Ms. Kelly also said “with respect” five times, a gentle way to either course-correct the conversation, or to ask a clarifying question of Mr. Trump).

“Mr. Trump, one of the things people love about you is they believe you tell it like it is,” Ms. Kelly said at one point. “But time and time again in this campaign, you have actually told the voters one thing, only to reverse yourself within weeks or even sometimes days.”

Fox News then reeled off three videos — on the war in Afghanistan, Syrian refugees and whether Mr. Trump thinks former President George W. Bush lied about the reasons he gave for the war in Iraq. It set off an intense 12-minute back and forth.

Mr. Rubio got involved, and brought up what he called the Better Business Bureau’s D-minus grade of Trump University in a review. Mr. Trump disputed this before Ms. Kelly intervened and confirmed “the rating from the Better Business Bureau was a D minus.” She then asked further questions about a lawsuit against Trump University and brought up yet another slide from a court opinion on the case.

Mr. Trump scoffed but did not express the same level of pushback as in the first debate.

After the debate on Thursday, in an interview with Bill O’Reilly of Fox News, he said he found the moderators to be “very fair.”

Mr. Trump later appeared with Sean Hannity of Fox News and gave Ms. Kelly and Fox News strong reviews.

“I really think the moderators were very, very good tonight, the three moderators,” he said. “I thought Fox did a very good job tonight.”

For the time being, détente.