"There was a picture on the front page of the National Enquirer, which does have credibility and they are not going to do pictures like that because they get sued for a lot of money if things are wrong, OK?” Trump said at a morning press conference in Cleveland.
The GOP presidential nominee added that he merely "pointed out" that the picture was published and that he did not know nor imply that Cruz's dad had anything to do with the assassination of John F. Kennedy.
"So they have a picture, and old picture having breakfast with Lee Harvey Oswald. Now, I'm not saying anything," said Trump. "This has nothing to do with me. Except I might have pointed it out. But it had nothing to do with me, I have no control over anything. I might have pointed it out.
"So here is the story. The press takes that and they say 'Donald Trump and his conspiracy theories, he went out and said that his father was with Lee Harvey Oswald and he assassinated the president.' What did I do?" he added.
Cruz, a Texas Republican and former presidential candidate, on Thursday defended his non-endorsement of Trump, stating that the GOP needs a candidate that does not "scream and yell and hurl insults."