Mr Trump pointed to the fact that during the caucus the Cruz campaign told voters rival Ben Carson planned to quit the race, which was not true.
The Cruz campaign apologised to Carson's camp, saying it was a misunderstanding.
Mr Trump also cited Cruz leaflets that accused Iowans of "voting violations".
Paul Pate, the top election official in Iowa, condemned the mailers on Saturday, saying the state does not track or grade individual voters.
Mr Pate said Mr Cruz's mailers "misrepresent Iowa election law" and that they were "not in keeping in the spirit of the Iowa Caucuses," but he stopped short of any official action.
"Ted Cruz didn't win Iowa, he stole it. That is why all of the polls were so wrong and why he got far more votes than anticipated. Bad!" Mr Trump said.
Earlier, he wrote on Twitter that Mr Cruz "illegally" won the caucus, but later deleted the tweet.
The reaction contrasts with his concession speech on Monday night, which was seen as a humble departure from his usual bombastic style.
Mr Trump placed second in Monday's contest, which he called "a long-shot great finish" in an earlier tweet.
Mr Cruz's camp is not taking the accusations too seriously.
"Reality just hit the reality star - he lost Iowa and now nobody is talking about him, so he's popping off on Twitter," Mr Cruz's communications director Rick Tyler told Politico in an email.
"There are support groups for Twitter addiction, perhaps he should find his local chapter."
There is no precedent for re-doing a caucus.
The remaining candidates are now in New Hampshire, the next state to hold a primary vote, where Mr Trump is leading in the polls.
Rand Paul, a Republican who represents the Libertarian wing on the party, dropped out of the race on Wednesday after finishing fifth in Iowa.
Presidential candidate Donald Trump has called for a new election in Iowa, accusing the Republican winner, Ted Cruz, of fraud.