McIlroy has been the biggest supporter of the established tours in their battle with the Saudi-funded breakaway circuit, but recently called for a compromise to be reached between the rival factions.
However, he believes that will be impossible if Norman remains at the helm of LIV Golf, which is expanding to a 14-event league schedule in 2023.
"I think Greg needs to go. I think he just needs to exit stage left," McIlroy told a pre-tournament press conference ahead of the DP World Tour Championship in Dubai.
"He's made his mark but I think now is the right time to say you've got this thing off the ground but no-one's going to talk unless there's an adult in the room that can actually try to mend fences."
LIV Golf last week denied reports that Norman would be replaced by former TaylorMade executive Mark King, while there is also the small matter of various lawsuits to be resolved before any compromise can be reached.
"There are obviously two lawsuits going on at the minute," McIlroy added. "There's the PGA Tour versus LIV and there's obviously this one that's coming up with the DP World Tour in February.
"Nothing can happen if those two things are going on. Right now it is a bit of a stalemate."
February's legal hearing will determine whether LIV players such as Ian Poulter and Lee Westwood can be banned from the DP World Tour and therefore ineligible to qualify for the Ryder Cup.
McIlroy admits his relationship with a number of Ryder Cup team-mates has been significantly damaged by their defection to LIV Golf and feels they should not be on Luke Donald's side in Rome next year.
But asked if "forgiveness" was possible in an increasingly polarised environment, McIlroy said: "Yeah, I think in time, that can happen.
"It's obviously been a very contentious year in golf. The best thing in golf is to have all the best players playing together, and (with) what's happening right now, that's not happening.
"So I fear for the game when that's going on. It's contentious because there's lawsuits going on and people suing people; it's very, very messy.
"If all that stuff can be sorted out one way or the other, then you can get to the stage where there's forgiveness and people can have dialogue and come to some sort of common ground or compromise.
"I don't think (golf) will have another year like this one because all of the noise has been about who is jumping ship, who is going where, who is staying, who is going.
"Very little of the storyline has actually been about the golf. So I think next year, if we can get the storylines to being about the golf and what's happening on the course, that's a good thing."