Carragher is facing the possibility of losing his job with Sky Sports after the unsavoury incident was caught on film.
The former Liverpool defender claimed that he had been taunted several times by the man and his daughter during the road traffic incident, but made a public apology on Sunday night.
Barton has slammed Carragher for his actions, and the former England international believes that the football pundit will now lose his role with Sky Sports.
"It's absolute madness, you just can't do that in any fashion. If Jamie is that annoyed he should just keep his window up, or just tell him to mind his own business," Barton told talkSPORT.
"I've done a lot of stupid stuff, and we all get stick, but to do that. Surely he's got to see the fact there's a young girl in the car... he's got kids himself! Why is he that emotionally attached to the game? He should be an impartial pundit.
"Look, he's made a mistake but say that was me, I think my position should rightly be under fire and I think his will be. It's down to Sky, but Sky would have a large subscription base of Manchester United fans, and also fathers and mothers – would they want to keep paying if he stays?"
Carragher retired from professional football at the end of the 2012-13 campaign.