Editor's note: This story has been updated throughout.

CLEVELAND — A day after formally accepting his party's presidential nomination, Donald Trump breathed new life into old battles against a long-vanquished Republican rival: U.S. Sen. Ted Cruz of Texas.

Addressing supporters Friday morning in Cleveland, Trump insisted he does not want Cruz's endorsement and re-litigated two of their nastiest clashes from the primaries, both involving members of the Texas senator's family. Trump's remarks came on the heels of a Republican National Convention at which Cruz pointedly declined to support Trump.

"I don't want his endorsement," Trump said Friday. "If he gives it, I will not accept it."

Read More How Ted Cruz Crashed Donald Trump's Convention

"Honestly, he should’ve done it because nobody cares and he would’ve been in better shape four years from now if he’s” going to run for president again, Trump added, going on to float the idea of setting up a super PAC to oppose Cruz in 2020.

Trump also provided a lengthy defense of his injection of Cruz's wife, Heidi, and father, Rafael, into the primaries this year, retweeting an unflattering photo of Heidi Cruz and giving voice to a conspiracy theory that Rafael Cruz was involved in the assassination of former President John F. Kennedy. Cruz said Thursday that a GOP unity pledge was "abrogated" when Trump dragged his family into the race.

“ Honestly, he may have ruined his political career. I feel so bad for him. ” — Donald Trump, on U.S. Sen. Ted Cruz

On the retweet of the Heidi Cruz photo, Trump again argued he did it because a super PAC that Cruz is “very friendly with” had put out an ad featuring a risqué picture of Trump’s wife, Melania, a former model. Cruz’s campaign had no connection to the super PAC, an anti-Trump outfit, and Cruz had denounced the ad as inappropriate.

“OK, so they sent out the first picture,” Trump said of the anti-Trump group. “Please remember that.”

Trump made clear he harbors no ill will toward Heidi Cruz, whom he called “a very nice woman and a very beautiful woman.” He then offered a backhanded compliment to her husband.

"I think it’s the best thing he’s got going — and his kids — if you want to know the truth,” Trump said. “In a certain way, although he’s got good intellect, but he doesn’t know how to use it, and he was a good debater but he didn’t do well in the debates against me.”

Trump insisted he "had nothing to do with" a story promoting the Rafael Cruz conspiracy theory in the National Enquirer, whose work he argued is worthy of Pulitzer prizes. Trump's only involvement in the story, he insisted, is that he "might have pointed it out."

"All I did is 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," Trump said, stirring the pot even more. "Now, Ted never denied that it was his father."

The GOP convention here, which ended Thursday night, put on full display the lingering rift between the nominee and Cruz. The Texas senator offered no support for Trump in a primetime address Wednesday, causing an angry reaction from delegates inside the hall — and even from some of his own constituents at a Texas delegation breakfast the following morning.

"Honestly, he may have ruined his political career," Trump said Friday. "I feel so bad for him."

"He’ll come and endorse because he has no choice, but I don’t want his endorsement," Trump added. "What difference does it make? I don’t want his endorsement. Just, Ted, stay home, relax, enjoy yourself."