United were outplayed at the Stadio Giuseppe Sinigaglia, extending their winless run to seven matches in all competitions and increasing the pressure on Carrick.
"I have not given my own position any thought," Carrick said when asked whether he feared being dismissed.
United fell behind inside two minutes when Nico Paz scored for Como, following their first-minute concession in last week's 2-0 defeat to Atletico Madrid.
Cesc Fabregas's side dominated and raced into a 4-0 lead before substitute Joshua Zirkzee scored a late consolation for United.
Carrick was angry that his players failed to build on their 1-1 draw against Leeds, which had offered signs of improvement.
"For us to play in that way, it's not acceptable," he said. "It wasn't a good performance at all."
He added: "That bit of intensity, that spirit, that aggression, that real willingness to give everything. Tonight was the total opposite performance. I have to say, it was tough to watch."
The United manager admitted he struggled to understand the dramatic difference between the two performances.
"Genuinely, it's difficult to understand how it can be so different," Carrick said, insisting that although poor displays happen, the manner of Wednesday's defeat was unacceptable.
Carrick also played down the significance of his post-match acknowledgement of the travelling supporters, saying he regularly thanks them for their backing.
United must now regroup before Sunday's Premier League meeting with Bournemouth at Old Trafford.
"I'm frustrated, more than frustrated, I'm really angry to be honest," Carrick said, admitting his team had considerable work to do before their next match.
For Como, the victory was a landmark moment in their first European campaign after finishing fourth in Serie A last season.
Fabregas has helped transform the club in three years, guiding them from the second division to competing among Europe's leading teams.
However, the former Arsenal and Barcelona midfielder warned his players against becoming carried away by their impressive victory.
"We must have a mentality to go out and do well in every match," Fabregas said. "I understand that beating Manchester United has more value, for the media, than normal."
Como face Sassuolo in Serie A on Saturday, and Fabregas stressed that his side must maintain their standards.
"So stay calm, humble, feet on the ground. The path does not change. This match doesn't change anything. We are at the start of everything," he said.
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