Over 8 million watched Saturday's event in Des Moines, Iowa.
Democratic candidates Hillary Clinton, Bernie Sanders and Martin O'Malley will face off in Iowa at 9 p.m. ET.
Carson's remarks came after the real estate mogul went on a lengthy rant about him.
GOP presidential candidates will face off in their fourth debate of the election season.
Republican candidates are seeking control over how the debates are run.
The new survey shows that voters are attracted to Carson's personal story.
The retired neurosurgeon leads the real estate mogul by 8 percentage points in the early primary state.
Fiorina had polled poorly in July and early August, but she gained support after the first Republican debate on August 6.
The debate, which is the third GOP event of this election cycle, will be held on October 28.
CNN would allow vice president to take part even if he declares his campaign the day of the event.
The real estate billionaire leads the retired neurosurgeon by one percentage point.
Despite an impressive showing at the last Republican debate, Carly Fiorina trails Ben Carson.
National support for Republican candidate Carson has reached 23 percent, just 4 percentage points behind Trump.
The CNN Republican presidential debate will take place in southern California on Wednesday.
Jim Gilmore could be left on the sidelines.
The top 10 Republican candidates in the polls will participate in the Thursday night debate on Fox.
Opinion polls after the event all produced different winners.
David Cameron reiterated his refusal to attend the debates, despite broadcasters saying they would go ahead without him.
Cameron added that he would be "happy for there to be no debates at all".
Charlie Crist brings a portable fan with him wherever he goes, and that's something Rick Scott wasn't a fan of on Wednesday.