Trump turned a conversation on "Fox & Friends" to mentioning a rumor reported in a story that appeared in the supermarket tabloid, The National Enquirer, that Cruz's father was connected to the assassin.
CNN host Wolf Blitzer said to Trump on Wednesday that he wanted him to clarify his previous claim.
"You don't really believe that Ted Cruz's father had anything to do with the assassination of President Kennedy," Blitzer said to Trump.
"No I don't," Trump replied. "Of course I don't think that."
Cruz dropped out of the race after Trump won in the Indiana primary on Tuesday. Blitzer told Trump that he is now the presumptive GOP nominee and Trump argued back that he wasn't when he circulated the Enquirer's story.
Trump said that Cruz's father said "horrible" things about him and that it's not a "one-way street" when he does something.
"So I was not a presumptive winner at that time. I was going against them, they were going against me," Trump said.
"But — bottom line you don't believe in that conspiracy," Blitzer countered.
"Of course I don't believe that. I wouldn't believe it, but I did say 'let people read it,'" Trump replied.