Washington, Feb 4  A day after his "humbling" second place finish in the first presidential nominating contests, Republican frontrunner Donald Trump was back in form as he accused his vanquisher Ted Cruz of stealing the Iowa caucus.

Attacking the arch conservative Texas senator Cruz in a series of tweets, the brash billionaire questioned the outcome of Monday's Iowa caucus that kicked off the 2016 presidential race and called for either new voting there or the results to be nullified.

"Based on the fraud committed by Senator Ted Cruz during the Iowa Caucus, either a new election should take place or Cruz results nullified," he wrote.

"Ted Cruz didn't win Iowa, he stole it," Trump tweeted. "That is why all of the polls were so wrong and why he got far more votes than anticipated."

Cruz's campaign dismissed Trump's claims on Twitter with humour.

"Reality hit the reality TV star in Iowa, so nobody is talking about him now, so he's trying to regain some attention on Twitter," Cruz's Communications Director Rick Tyler told CNN.

"There are Twitter addiction support groups, so he should seek out his local chapter."

Cruz himself teased Trump on Twitter on Wednesday, suggesting his rival become a Democrat and linking to one of his anti-Trump ads.

"Bernie is contesting #IA results. Maybe Donald should go back to IA & join the Ds. Bet they'd love #TrumpCare," Cruz tweeted.

Cruz then added, "Yet another #Trumpertantrum... @realDonaldTrump very angry w/the people of Iowa. They actually looked at his record."

Polls in the week leading up to the caucuses showed Trump five points ahead of Cruz, which raised expectations for Trump's campaign, but Cruz finished first with 28 percent support, four points more than Trump.

Trump also slammed Cruz for sending out deceitful mailers and suggesting in a release that retired neurosurgeon Ben Carson was quitting the race.

Trump tweeted that "during primetime of the Iowa Caucus, Cruz put out a release that @RealBenCarson was quitting the race, and to caucus (or vote) for Cruz."

"Many people voted for Cruz over Carson because of this Cruz fraud," Trump said of the Cruz campaign's caucus day actions.

Meanwhile, Ben Carson hit CNN for what he said was an inaccurate report that he was leaving the race on Monday.

The network said it had only reported that Carson would go to Florida instead of New Hampshire or South Carolina immediately after the caucuses to take a "deep breath" and get clean clothes.