Carson said the Texas senator -- who won the Iowa GOP caucus with 27.6 percent of the vote -- apologized to him after his staff mistakenly told Iowans Monday night that Carson was quitting the race.
But the retired neurosurgeon was clearly not satisfied with an apology alone, saying the mistake contributed to his fourth-place finish in Iowa with 9.3 percent of the vote. But Carson declined to criticize Cruz directly, suggesting the senator should fire the individuals responsible for the trouble.
"When I discovered there were things in my campaign that I couldn't agree with after really doing an investigation, I made changes. I think that is what a good leader does," Carson said at the National Press Club.
An adviser to Carson told the media Monday night that Carson would not immediately go on to New Hampshire or South Carolina, but instead fly to Florida to get a "fresh set of clothes."
Carson disowned this comment, which some interpreted as a sign the campaign was winding down.
"I didn't make that announcement," Carson said. "I didn't say it so don't blame me."
Cruz said in a statement Tuesday that his campaign should have clarified that Carson was not dropping out of the race when that fact became clear.
"Last night, when our political team saw the CNN post saying that Dr. Carson was not carrying on to New Hampshire and South Carolina, our campaign updated grassroots leaders just as we would with any breaking news story," Cruz's statement read.
"That's fair game. What the team then should have done was send around the follow-up statement from the Carson campaign clarifying that he was indeed staying in the race when that came out."
Donald Trump piled on to the controversy, saying Cruz committed "fraud" and calling for a new election in Iowa.
"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!" Trump tweeted Wednesday.
"Many people voted for Cruz over Carson because of this Cruz fraud. Also, Cruz sent out a VOTER VIOLATION certificate to thousands of voters," Trump added.