Pressure piled on the under-fire manager, who has overseen a run of seven games without a victory, which began with United’s 1-0 derby defeat by Manchester City.
They were torn apart by Cesc Fabregas’ men in the Champions League on Wednesday, to follow on from their dismal 2-0 loss at Atletico Madrid last time out, which prompted Lisandro Martinez to openly take aim at the club’s hierarchy.
They were blown away at the Stadio Giuseppe Sinigaglia, as Nico Paz starred with two goals.
"It was more than disappointing,” Carrick told TNT Sports.
“I think everybody has got to take responsibility for that. I 100 per cent take responsibility for the overall team and the performance - that's where it stops.
“It stops with me and I'm absolutely fine with that. I accept that totally. But as a supporter of this football club for so long, to see a performance like that is unacceptable, I have to say.
"I care an unbelievable amount about the supporters. They are the heartbeat of this football club and they always will be. For them to travel, I had to apologise for the performance. To have to watch a performance like that, I totally feel for them.
“There are no excuses at all. It is not acceptable."
Carrick, who took over the reins from Ruben Amorim on a full-time basis after a successful second spell as interim head coach, would not be drawn on whether he had the backing of the United hierarchy, insisting that now was the time to “digest” the game.
The Red Devils return to Premier League action on Sunday when they welcome Bournemouth to Old Trafford.