Donald Trump, angry at a Facebook campaign showing a cheesecake photo of his wife, threatened to retaliate against the reputation of Heidi Cruz in multiple tweets Tuesday night.

In the first tweet, Mr. Trump berated rival Sen. Ted Cruz of Texas, saying “that is some low level ad you did using a picture Melania in a GQ shoot. Be careful or I will spill the beans on your wife.”

Mr. Trump quickly deleted that tweet, though not before it had been preserved for history by numerous political reporters.

Despite the unusually personal nature of the attack, neither Mr. Cruz nor his wife had anything to do with the ad from the Make America Awesome PAC, an anti-Trump super PAC that is legally barred from any ties to Mr. Cruz or his campaign.

But then Mr. Trump doubled down, adding his epithet for Mr. Cruz, though he did not tag the senator’s own Twitter handle this time.

“Lyin’ Ted Cruz just used a picture of Melania from a G.Q. shoot in his ad. Be careful, Lyin’ Ted, or I will spill the beans on your wife!” he said in the second tweet.

In neither tweet did he specify what he would “spill the beans” about.

In reply, Mr. Cruz took to Twitter to point out that he had nothing to do with the picture and calling Mr. Trump classless and a coward.

“Pic of your wife not from us. Donald, if you try to attack Heidi, you’re more of a coward than I thought. #classless,” Mr. Cruz replied.

The real-estate tycoon and Republican front-runner was irked by a Facebook campaign by an anti-Trump group seeking to drive up turnout among conservative Mormon women in Utah and Arizona.

The Make America Awesome PAC distributed a picture of Mrs. Trump, a former professional model, curled up on a fur rug with no clothing visible. The risque shot is accompanied by the text “Meet Melania Trump. Your next first lady. Or you could support Ted Cruz on Tuesday.”

Liz Mair, a strategist with Make America Awesome, also noted on Twitter that the picture wasn’t being circulated by Mr. Cruz’s campaign.

“Hi Donald, I know you’re really upset about that ad, but it was Make America Awesome’s, not Ted Cruz‘s,” she wrote.

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