Editor's note: This story has been updated throughout.

Battling the perception he is running a dishonest campaign, Ted Cruz on Monday sought a major change, asking for the resignation of his top spokesman.

It was not immediately clear if the spokesman, Rick Tyler, had heeded Cruz's request. A spokeswoman for the senator said late Monday that Tyler was no longer with the campaign.

Cruz's move comes as he faces persistent questions from his GOP rivals about the integrity of his campaign. Earlier Monday, Tyler publicly apologized after sharing a video online that falsely purported to show U.S. Sen. Marco Rubio of Florida making a negative comment about the Bible.

"I've spent this morning investigating what happened, and this morning I asked for Rick Tyler's resignation," Cruz told reporters in Las Vegas, calling what Tyler did a "grave error of judgment."

Rubio's campaign responded to the news by praising Tyler's communication skills — but doubling down on its criticism of his boss as a unethical candidate.

"Rick is a really good spokesman who had the unenviable task of working for a candidate willing to do or say anything to get elected," Rubio spokesman Alex Conant said in a statement. "There is a culture in the Cruz campaign, from top to bottom, that no lie is too big and no trick too dirty. Rick did the right thing by apologizing to Marco. It's high time for Ted Cruz to do the right thing and stop the lies."

The controversy stemmed from a video that surfaced over the weekend that portrayed Rubio making a dismissive remark about the Bible as he passed a Cruz staffer and Rafael Cruz, the candidate's father, in a hotel lobby. In the subtitles of the initial version of the video, Rubio says there are "not many answers" in the Bible.

The Rubio campaign insisted the transcript of the video was inaccurate, saying the Florida senator had said "all the answers" are in the Bible. The original source of the video, the University of Pennsylvania's student newspaper, later said the audio was "too unclear to say for sure" what was said.

Over the weekend, Tyler had linked on Twitter to a story showing the video. He issued an apology late Sunday, saying he had since learned from the Cruz staffer that Rubio had "made a friendly and appropriate remark."

"The story misquoted a remark the Senator made to the staffer," Tyler wrote on Facebook. "I assumed wrongly that the story was correct."

Rubio had demanded accountability for Tyler's action while speaking with reporters Monday in Las Vegas.

"Who's going to be fired for making up this video?" Rubio asked, a calling it part of a "very disturbing pattern of deceptive campaigns and flat-out just lying to voters."

Billionaire Donald Trump, who has also accused Cruz of running a dishonest campaign, quickly reacted to Cruz asking Tyler to resign, tweeting that Cruz "falsely suggested Marco Rubio mocked the Bible and was just forced to fire his Communications Director. More dirty tricks!"

Tyler is by far the highest-profile member of Cruz's communications team, a regular on cable news with a reputation for his quick and caustic wit. But Tyler, who previously worked for Newt Gingrich when he ran for president in 2012, had put the Cruz campaign in an awkward position on more than one occasion, including when he criticized Sarah Palin ahead of her endorsement of Donald Trump. Cruz later said he held no ill will against Palin, a former ally, for backing the billionaire.

The staff shakeup seems to be an acknowledgment that efforts by Rubio and Trump to brand Cruz a liar are making an impression on the Texas senator's campaign. The criticism began in the hours after the Iowa caucuses, when Ben Carson's campaign accused Cruz's team of falsely telling supporters the retired neurosurgeon was dropping out of the race. The drumbeat has apparently taken a toll: A third of Republican voters in South Carolina thought Cruz ran the most unfair campaign there, second only to Trump, according to exit polling done on the Saturday primary.

Cruz's call for Tyler's resignation highlighted a rough day for the Cruz campaign that began with prominent allies publicly questioning its direction after a disappointing third-place finish in Iowa. Amanda Carpenter, a former top staffer for Cruz on Capitol Hill, tweeted that Cruz needs to "go back to what worked in his TX Senate race" and reclaim the mantle of anti-Washington crusader. Steve Deace, one of Cruz's biggest backers in Iowa, also took to Twitter to offer some advice to the campaign, urging Cruz to stop attacking Rubio and focus all his firepower on Trump.

Dave Weigel with the Washington Post contributed to this report.