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.
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