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The next Republican debate, originally scheduled to be hosted by Fox News on Monday in Salt Lake City, has been canceled, the network confirmed on Wednesday.

The news comes after Donald J. Trump announced that he would skip the debate, and Gov. John Kasich of Ohio said he would not attend if Mr. Trump would not either.

“This morning, Donald Trump announced he would not be participating in the debate,” Michael Clemente, the executive vice president of news at the network, said in a statement. “Shortly afterward, John Kasich’s campaign announced that without Trump at the debate, Kasich would not participate. Ted Cruz has expressed a willingness to debate Trump or Kasich — or both. But obviously, there needs to be more than one participant. So the Salt Lake City debate is canceled.”

Mr. Trump said during an interview on the “Fox and Friends” program earlier on Wednesday that he would not attend.

“How many times can the same people ask you the same question?” Mr. Trump said in the interview. “So I was very surprised when I heard that Fox called for the debate. Nobody told me about it, and I will not be there, no.”

He cited a previous commitment, a speech to the American Israel Public Affairs Committee — “a major speech in front of a very important group of people” — as his reason for not doing the debate.

His announcement follows a tantrum, of sorts, that the candidate conducted on Twitter on Tuesday night, unleashing once again a string of criticisms at Megyn Kelly, the Fox News host and debate moderator with whom he has made disparaging comments in the past. (Mr. Trump and Ms. Kelly were pleasant enough to each other, for the most part, at the Fox debate on March 3.)

But Mr. Trump had wavered about attending Monday’s event beforehand, stating in a news conference on Friday that he thought there had been enough debates, but also telling Chris Cuomo on CNN after the last debate that “I think they want to do two more debates, and I guess I’m pretty much O.K. with it.”

With Mr. Trump announcing he won’t be there, John Weaver, the chief strategist for Mr. Kasich’s campaign, announced on Twitter that the Ohio governor would not be there, either.

No debate in SLC Monday due to Trump backing out. No surprise Trump avoiding contrast. Utah: @JohnKasich headed your way Fri. — John Weaver (@JWGOP) March 16, 2016

He did say if Mr. Trump changes his mind, then Mr. Kasich would attend. “If Trump changes his mind — as Carson said there are 2 Trumps — we will be there,” Mr. Weaver wrote. “Positive contrasts nicely with division.”

Mr. Cruz took Mr. Trump to task, resurrecting his “Ducking Donald” criticism he used when Mr. Trump skipped an earlier Fox News debate in Iowa.

“Donald is scared to debate because he knows Ted Cruz will expose him as unprepared to be president and commander in chief,” a page on the Texas senator’s campaign website reads.