The flap began when Trump called out a Facebook ad targeting conservative Mormon voters, showing Trump's wife, Melania, posing nude for a GQ photoshoot. Trump vented his anger at Cruz, though the ad itself was created by an anti-Trump super PAC, not the Cruz campaign.
Trump then tweeted he would "spill the beans" about Heidi Cruz, a top executive at Goldman Sachs. It was not clear what Trump was referring to with the threat.
Later Wednesday, Trump retweeted a supporter's side-by-side comparison of Heidi Cruz and Melania Trump with the note "a picture is worth a thousand words."
The scuffle has drawn wide criticism. A CNN anchor grilled a Trump adviser over the matter.
Fox News host Megyn Kelly, who has herself been at the receiving end of Trump's anger on Twitter, issued a one-word response to the photo of the two women.
"Seriously?" Kelly asked.