Rick Tyler worked as the Communications Director for Mr Cruz when he posted a video online with incorrect subtitles suggesting Mr Rubio was criticising the Bible.
Mr Cruz told reporters in Las Vegas that he had asked him to step down in order to "conduct this campaign with the very highest standards of integrity".
Responding in a statement on Facebook, Mr Tyler said: "I want to apologize to Senator Marco Rubio for posting an inaccurate story about him here earlier today.
"I assumed wrongly that the story was correct. 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."
It is not the first gaffe by the Cruz campaign. Last month he had to apologise after it emerged some of his campaign team had spread rumours about another Republican candidate, Ben Carson suggesting he was withdrawing from the race.
Ted Cruz currently trails Donald Trump in the race to win the Republican candidacy for the presidential election later in the year.