Carrick's position is under mounting pressure after a poor showing in the Red Devils' 4-1 UEFA Champions League defeat at Como.
That means United have won two of their 11 games across all competitions so far this season, with just one victory in the Premier League.
As part of the post-match fall out, Fernandes called out Carrick's tactics, and labelled United's approach to pressing Como as 'lacking bravery', with confusion over what the plan was.
Carrick's response centred on a 'lack of running' from his players and Carragher has slammed Fernandes' stance.
"For Fernandes to do that, it's a disgrace. Last season he was Player of the Year. That was due to Michael Carrick coming in as manager. He wouldn't have got that award without Carrick. He put him back in his right position.
"His attitude and the way he goes about things, it's not right. He's reacted like that because he was brought off and he's been embarrassed by the manager.
"He talks about pressing, whether it's man-to-man or zonal, for years, Bruno Fernandes has meant United can't press.
"He runs all over the place like a kid in the school yard for the supporters. They think he's working hard, he's not. He's killing the rest of the team by doing it.
"It's a disgrace that he's called out the manager considering how he's performed this season. Awful attitude and not what a Manchester United captain should be."