"If he gives it, I will not accept it," Trump said at a news conference in Cleveland at the close of the Republican National Convention.
"I don't want his endorsement," he added. "Just, Ted, stay home, relax, enjoy yourself."
He also suggested that if Cruz were to seek the White House, he would set up a super PAC to oppose him.
And Trump, speaking the end of a week dedicated to unifying the party, once again revived the conspiracy theory published in the National Enquirer that linked Cruz's father to John F. Kennedy assassin Lee Harvey Oswald -- an accusation that has no evidence behind it.
"I don't know his father. I met him once," Trump said. "I think he's a lovely guy, a lovely guy. All I did was point out the fact that on the cover of the National Enquirer, there was a picture of him and crazy Lee Harvey Oswald having breakfast. Now Ted never denied that it was his father."
Trump first suggested that Cruz's father was involved in the assassination of Kennedy the morning of the Indiana primary, citing the tabloid story.
Cruz allies have since repeatedly pointed to that low blow from Trump as a reason for refusing to support the Republican nominee. And on Thursday, Cruz referenced the incident, as he drew an angry reaction from members of his state's own GOP delegation.
"I am not in the habit of supporting people who attack my wife and attack my father," Cruz said.
Trump devoted a significant portion of Friday's event, billed as a thank you to convention staff and organizers, to attacking his former primary rival.
Speaking at the RNC on Wednesday, Cruz had declined to endorse Trump, instead urging delegates to vote their "conscience."
Cruz said that the Trump campaign had seen the speech beforehand, and that he told Trump personally that he would not make an endorsement.
But Trump appeared to dispute that on Friday morning, saying Cruz went off-script at the convention.
"So Ted Cruz took his speech that was done, was on the Teleprompter, said hello, then made a statement that wasn't on the speech, then went back to his speech," Trump said. "See, to me, that's dishonorable."
Trump likewise said it was "dishonorable" of Cruz to abandon his pledge to support the Republican nominee.
But Trump said that Cruz's camp started it with an ad that ran in Utah showing racy photos of his wife, Melania Trump. The ad was put out by a PAC with no official ties to the Cruz campaign, but Trump said Friday that he doesn't buy that.
"Folks, a lot of us are political people," Trump said. "We're not babies. His people are on the PAC."
At one point in his remarks, Trump even suggested forming his own PAC to oppose Cruz if the senator runs again in four years.
"I don't see him winning anyway, frankly. But if he did, it's fine," Trump said. "Although maybe I'll set up a super PAC if he decides to run."
Turning to his running mate, Indiana Gov. Mike Pence, Trump asked, "Are you allowed to set up a super PAC, Mike, if you are the president to fight somebody?"