Trump sent out a flurry of tweets Wednesday morning calling for the caucus results to be invalidated, and threatened to sue.
"It is no surprise that Donald is throwing yet another temper tantrum or if you like, another Trumpertantrum," Cruz said.
"We need a commander-in-chief, not a twitterer-in-chief," Cruz said, noting that Trump finds it hard to lose.
The Texas Republican said his daughters, who are 5 and 7, are better behaved than the real estate mogul.
"Caroline and Catherine are better behaved than a presidential candidate who responds by insulting everyone every day when he loses," Cruz said.
Cruz said he is amused by Trump's tweets.
"I think they’re very funny," Cruz said. "I wake up every day and laugh at the latest thing Donald has tweeted because he’s losing it."
Trump skipped the Republican debate in Iowa last week, causing Cruz to challenge him to a 'mano a mano' debate. Trump didn't take him up on the offer. Cruz put out another offer Wednesday, saying that Trump can stand there and insult Cruz, who will only talk about issues of substance. Cruz said the businessman is only lashing out because he lost the Iowa caucuses.
"When he loses, he blames everybody else. It’s never Donald’s fault. It’s always somebody else’s fault and it’s always an insult," Cruz said.