The Dundee-born star said he should have "thought better" after accepting the honour in 2003 for his services to drama.
Cox, 73, has campaigned for Scottish independence and said he didn't think he would be made a Sir because of that.
But he insisted he would turn it down anyway as he feels uncomfortable with the royal family and the honours system.
He said: "When I became a CBE I did it at a time when I should have thought better but I accepted it because I'm fickle.
"I have a royalist sister and a republican sister and I thought well my royalist sister will be happy but she couldn't give a s**t. That was my justification.
"Now I would never be a knight. I wouldn't want to do that.
"A CBE is usually a precursor to getting a knighthood. I got my CBE a while ago and I have been so active in Scottish independence.
"They are determined to keep the Queen, she goes with the whole nine yards, we don't get rid of her. It will never be a proper republic."
Cox won best actor in a TV drama for his role as media magnate Logan Roy in Succession at the Golden Globe awards earlier this month.
Speaking on the Marc Maron WTF podcast, he also told how fans now ask him to tell them to 'F**k off' in the style of the foul-mouthed character.
He added: "I come out of the stage door at the theatre and people come up to me and say 'Could you tell us to f**k off please, can I film you saying f**k off?'.
"The funniest was I went to a Me Too event and all these women were coming up to me and saying 'Could you tell me to f**k off?'
"I'm going 'Is this wholly appropriate at a Me Too event?'."
Earlier this month, New York-based Cox caused controversy after he confessed he is a regular cannabis smoker and urged others to try the drug.
He said he used the drug to avoid being square and to help him cope with his frustration at modern politics.
Cox said he didn't try cannabis until he was 50 as a result of feeling he had missed out on experimenting as a youth because of his devotion to acting.
His comments saw him dropped as the patron of a daycare centre for pensioners in Dundee.
The Mid-Lin Daycare Centre said in light of Dundee’s well-documented drug problems, they could no longer be associated with the actor.