The tweets are a marked contrast from the morning after the Iowa vote, in which Mr Trump offered his congratulations to his rival.
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Trump still has a double-digit lead in New Hampshire, though Cruz is in second and the Iowa win is expected to boost his numbers.
Cruz had apologised to Carson on Tuesday because of an email from his campaign before the caucuses, which are the crucial first vote in the United States presidential nominating process.
Meanwhile, Ben Carson hit CNN for what he said was an inaccurate report that he was leaving the race on Monday.
Mr Rubio is trying to be the Republican establishment’s preferred alternative to Mr Trump or Mr Cruz.
“Many people voted for Cruz over Carson because of this Cruz fraud”, Trump said of the Cruz campaign’s caucus day actions.
Trump’s revised assessment of the Iowa outcome came as the still-large field of Republican candidates and two Democratic contenders fanned out across the northeastern state of New Hampshire, where the next voting in the presidential race is set for Tuesday.
The Iowa caucuses also saw an unexpected victory for Republican candidate Ted Cruz, who pulled ahead of frontrunner Donald Trump.
Donald Trump admitted that his choice to skip the Fox News-hosted GOP debate could have contributed to his second-place finish in the Iowa caucuses Monday night.
Place among the top three candidates ranked according to the popular vote in the Iowa Republican caucuses on February 1, 2016.
“I think we’re going to do well here. It could have been a little bit better, could have been one notch better”, he said at a press conference before the rally. Ohio Gov. John Kasich, former Florida Gov. Jeb Bush and New Jersey Gov. Chris Christie all hope New Hampshire will breathe life into their flagging campaigns.
The poll found the most support for Rubio in a theoretical three-man race for the nomination, with 34% picking the Florida senator compared with 33% for Trump and 25% for Cruz.
“For a guy known for making bombastic, off-the-wall speeches, Donald Trump’s Iowa speech was something genuinely new: a conventional concession speech”, Timothy B. Lee wrote on Vox.
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On Wednesday, though, he gave a cautious outlook on his prospects in New Hampshire and beyond.
