(CNN) Ben Carson charged on Tuesday that Ted Cruz needs to fire campaign staffers for falsely telling Iowa caucusgoers to support the Texas senator because the retired neurosurgeon was planning on quitting the race.

"If Ted Cruz doesn't know about this, then he clearly needs to very quickly get rid of some people in his organization," Carson said on Fox News' "Fox and Friends." "And if he does know about it, isn't this the exact kind of thing that the American people are tired of? Why would we want to continue with that kind of shenanigan?"

"If he did know about it, he needs to come out and admit what he did and try to offer a solution," Carson added.

Cruz issued an apology to Carson later on Tuesday calling the incident a "mistake."

Cruz said in a statement Tuesday that his campaign staff saw a CNN report that Carson was dropping out, although CNN had not characterized Carson's actions that way.

"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 said in a statement first shared with CNN. "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."

The Cruz campaign had seized on the news late Monday that Carson would not travel directly from Iowa to New Hampshire or South Carolina, and incorrectly suggested to precinct captains it was a sign that he would be dropping out of the race while the caucus was still in progress.

Cruz Iowa staffer Spence Rogers sent precinct captains an email during the caucus hinting that Carson planned to leave the race, despite the fact that Carson's campaign said he would continue.

"Breaking News. The press is reporting that Dr. Ben Carson is taking time off from the campaign trail after Iowa and making a big announcement next week," the email, acquired by CNN, read. "Please inform any Carson caucus goers of this news and urge them to caucus for Ted Cruz."

Meanwhile, Cruz surrogate Iowa Rep. Steve King told followers on Twitter that "Carson looks like he is out."

Carson looks like he is out. Iowans need to know before they vote. Most will go to Cruz, I hope. https://t.co/lW5Js50EMA — Steve King (@SteveKingIA) February 2, 2016

Carson told Fox News that his wife, Candy, was left to correct the Cruz campaign's misinformation at one event.

"She spoke at one of the caucus sites and before she got there one of the Cruz people had again disseminated that information to the crowd," he said. "And one of the organizers told her 'I'm so glad you're here, so you could set the record straight.' And she did and we actually won in that precinct."

Carson's campaign was furious when they first learned of the tactic, and Carson decried it as a "dirty trick" during remarks Monday night at a party with supporters in West Des Moines.

"This is really a dirty trick," Carson said. "This makes me more determined than ever to keep going."

Cruz's campaign defended the Rogers message to the precinct captains.

"Carson said he was taking time off to go home. We simply repeated what he said," Cruz spokeswoman Catherine Frazier said Monday. "Our email simply stated the facts."

When asked what the "big announcement" mentioned in the email referred to, Frazier did not respond.

Carson will go home to Florida following the Iowa caucuses instead of flying directly to early-voting states New Hampshire and South Carolina, regardless of what happens in the first-in-the-nation voting state, so he can "get a fresh set of clothes," his campaign said Monday.

Larry Ross, a spokesman for Carson, insisted in a statement that the retired neurosurgeon "is not suspending his presidential campaign, which is stronger than ever."

"After spending 18 consecutive days on the campaign trail, Dr. Carson needs to go home and get a fresh set of clothes," Ross said. "He will be departing Des Moines later tonight to avoid the snow storm and will be back on the trail Wednesday. We look forward to tonight's caucus results and to meaningful debates in New Hampshire and South Carolina."

Campaign manager Ed Brookover characterized the break as a "deep breath" after Iowa.

Carson will remain in Florida until he travels to the Washington, D.C., area for meetings on Wednesday. He'll attend the National Prayer Breakfast on Thursday, and then he'll head to North Carolina for a fundraiser and New Hampshire for Saturday's Republican debate.

Despite the unusual travel plan, Carson projected confidence as voters packed caucus sites across the Hawkeye State, telling CNN's Wolf Blitzer earlier on Monday that he believes there is "a very strong chance" he'll finish among the top three candidates.

"The people who support me are extremely enthusiastic," Carson said, suggesting pollsters have underestimated his backing.

Carson told Blitzer he would reassess his campaign no matter where he finished in Iowa.