The Kentucky senator is boycotting the undercard GOP debate after he was demoted from the prime-time stage.
Trump said he influenced the network's decision to cut the New Hampshire paper from the debate.
The highest-polling GOP presidential candidates lined up across the prime-time stage Tuesday night.
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.