"Meet Melania Trump, your next first lady", read the online ad, posted on Facebook by the group Make America Awesome.
CNN has no reporting to suggest that the tabloid allegations are true or that Trump or any of his advisers are behind the story, which was published this week.
"You know, Donald is fond of giving people nicknames, with this pattern he should not be surprised to see people calling him "Sleazy Donald" because that is his first and last redoubt, to turn to sleaze", said Cruz.
The Cruz campaign has denied involvement with the ad.
"Ted Cruz's problem with the National Enquirer is his and his alone, and while they were right about O.J. Simpson, John Edwards, and many others, I certainly hope they are not right about Lyin' Ted Cruz".
"Of course the National Enquirer story is 100% FALSE!!!" she tweeted".
"I'm sorry, I don't think that's moving on at all, Adriana", she said, as Carpenter uttered a single sarcastic laugh.
It's an escalation of the back-and-forth between the two.
Chip Somodevilla Getty Images Donald Trump left and Ted Cruz at a debate
After Heidi Cruz introduced her husband at a rally Thursday evening in Janesville, Wisconsin, he acknowledged she "has been in the news the past couple days".
"Our spouses and our children are off-bounds", Cruz told reporters while campaigning in Dane, Wis. "It is not acceptable for a big, loud, NY bully to attack my wife". Ted Cruz was right to defend his wife, but in no universe should this conversation have any place in a presidential primary. Donald, if you try to attack Heidi, you're more of a coward than I thought.
That message and others have prompted an outcry among Republicans and Democrats alike, and Cruz said Thursday that "real men don't bully women".
Earlier, Trump shared an image on Twitter comparing his wife, Melania, a former model, to Cruz's wife, stating, "A picture is worth a thousand words".
"If Donald wants to get in a character fight, he's better off sticking with me, because Heidi is way out of his league", Cruz said during an appearance on CNN, borrowing a line from the movie "The American President".
Trump issued a statement instead of a tweet, his usual form of communication with supporters.
You can't blame Cruz for seeing Trump's tiny fingerprints on the story-and it's fully plausible that the mogul or one of his allies (rather than a Rubio booster) gave the story to the tabloid.
The race turned personal on Wednesday when a super PAC (political action committee) backing Mr. Cruz used a photo of Mr. Trump's wife Melania naked in a magazine spread, prompting a sharp response from the front runner.
When asked how many Muslims there are in the US, Mr. Cruz admitted he did not know.