"I'm making a very major speech in front of a very important group of people," Trump said on "Fox & Friends," later confirming that the remarks will be delivered before the American Israel Public Affairs Committee (AIPAC). Trump said his speech at the powerful lobbying group's conference in Washington "was scheduled a while ago."
AIPAC invited all presidential candidates to speak, but only Trump and Democrat Hillary Clinton are listed as speakers so far.
Fox said it called off the debate after Trump and Kasich backed out.
"Ted Cruz has expressed a willingness to debate Trump or Kasich — or both," Fox News Channel Executive Vice President Michael Clemente said in a statement. "But obviously, there needs to be more than one participant. So the Salt Lake City debate is cancelled."
Cruz, the Republican senator from Texas, told Fox News on Wednesday night that Trump "doesn't like when anyone challenges him. He was saying just a week ago that he was eager to get one-on-one with me. Well, the field has narrowed; he could have had a direct debate with me.
"And yet Donald apparently is ducking; he's afraid of being challenged."
The debate would have been the 13th of this election cycle for Republicans; at the first, in August, Fox News anchor Megyn Kelly opened by questioning Trump about disparaging remarks he has made about women — sparking a feud that has simmered for seven months.
In January, Trump skipped a Fox debate in Des Moines, a move he later said may have contributed to his second-place finish in Iowa's first-in-the-nation caucuses.
"I think some people were disappointed that I didn't go," Trump said in early February.
He participated in a debate earlier this month in Detroit and avoided any confrontations with Kelly, though he has continued to disparage her.
After Trump told Fox that he wouldn't come to its next debate, some immediately cast doubt on his explanation, suggesting that a group such as AIPAC would allow a leading presidential candidate multiple opportunities to speak before its members.
Dan Senor, whose tweet is embedded above, served on Sen. Marco Rubio's National Security Advisory Council.
On Fox, Trump said he thought the Republican field had held "enough" debates.
"We've had 11 or 12 debates; I did really well on the last one, I think I've done well on all the debates," he said. "I mean, according to Drudge and everybody else, I won the debates. But I think we've had enough.
"How many times can the same people ask you the same question? So I was very surprised when I heard that Fox called for a debate. Nobody told me about it and I will not be there, no."
"You know who loves them?" Trump said of debates. "The networks love them."
This post has been updated.