Ted Cruz has fired a senior aide after a video was published which falsely accused rival Republican candidate Marco Rubio of doubting the Bible.

Rick Tyler, Cruz's communications director, was sacked after what the Texas senator described as a "grave error". Tyler had circulated a 21-second video showing Rubio walking past Rafael Cruz – Ted's father – and another member of the Cruz team reading the Bible.

According to the video Rubio said: "Got a good book there" but adds "not many answers in it." However what Rubio actually said was: "All the answers are in there."

"This morning I asked for Rick Tyler's resignation," Cruz told reporters in Las Vegas.

"This was a grave error of judgment," he said. "It turned out the news story he sent around was false. But I'll tell you, even if it was true, we are not a campaign that will question the faith of another candidate."

The conservative website The Right Scoop had originally posted the video, commenting: "Was he saying that because Rafael Cruz is Evangelical and he's Catholic? That seems like small justification to be so snide about the Bible. Good luck with Evangelicals after that, Marco."

However the blog was updated to include an admission the subtitles may be incorrect and Tyler posted an apology on his personal Facebook page:

"I assumed wrongly that the story was correct," he wrote. "According to the Cruz staffer, the Senator made a friendly and appropriate remark.

"Since the audio was unclear, I should not have assumed the story was correct. I've deleted the post because I would not knowingly post a false story. But the fact remains that I did post it when I should have checked its accuracy first. I regret the mistake."

Cruz, who has focused his campaign on winning the vote of evangelical Christians, said his campaign had been subject to personal attacks from rivals.

"None of us have heard me throw the kind of insults at Marco Rubio that he throws at me every single day. If other candidates choose to go into the gutter, we will not do the same," he said.

"They have a long record they've earned in South Carolina of engaging in this kind of trickery and impugning the integrity of whoever their opponent is to distract the attention. We are going to stay focused on issues and substance and record."

However Rubio spokesman Alex Conant said Tyler is "a really good spokesman who had the unenviable task of working for a candidate willing to do or say anything to get elected".

"There is a culture in the Cruz campaign, from top to bottom, that no lie is too big and no trick too dirty," said Conant. "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."