Earlier this week, an anti-Trump Super PAC published the ad featuring Trump's wife Melania lounging nude, and Trump threatened to "spill the beans" on Cruz's wife.

"Donald, real men don't attack women. Your wife is lovely, and Heidi is the love of my life," Cruz said in a post on Twitter early on Thursday.

His tweet followed one moments earlier by Trump, the Republican front-runner in the race, in which he retweeted an image featuring a less-than-flattering picture of Heidi Cruz juxtaposed to a glamorous photo of Melania.

The war of words comes on the heels of the candidates' split win earlier this week. Trump won Arizona and Cruz took Utah in the latest state primary contests to represent the party in the Nov. 8 presidential election.

Both candidates are trying to garner enough delegates to win the Republican party nomination ahead of the nominating convention this summer. After Tuesday's contests, Trump had 738 of the 1,237 delegates needed to win the nomination, according to The New York Times. Cruz had 463.

Ohio Governor John Kasich is also in the running, but so far has only won his home state.

Cruz, a U.S. senator from Texas, is a staunch social conservative and a divisive figure in the party due to his willingness to criticize the leadership and his prominent role in bringing about a 2013 government shutdown.

But he is still seen by party grandees and many Republicans in Congress as preferable to Trump, a real-estate billionaire viewed as straying even further from party principles.

Both candidates are trying to line up more support, including endorsements from the dozen or so candidates who have dropped out of the race, to nail down votes ahead of a potential showdown at the convention in Cleveland in July.

On Wednesday, Cruz won the backing of former rival Jeb Bush as prominent Republicans overcome their aversion to the conservative senator.

U.S. Senator Lindsey Graham and Carly Fiorina, who had sought the nomination, have also endorsed Cruz while Trump has won the backing of former candidates Chris Christie and Dr. Ben Carson.

On Thursday, Graham said Republicans are likely to lose the November election to Democrats, who are deciding between former Secretary of State Hillary Clinton and U.S. Senator Bernie Sanders.

"We're already split," he told MSNBC. "It's no longer about winning the election for me. It's about trying to salvage a party that I love and conservatism as I know it."

The next Republican contests will be on April 5 in Wisconsin and on April 9 in Wyoming.