Carrick labelled a 4-1 thrashing by Champions League debutants Como on Wednesday, which stretched United's winless run to seven games, as "totally unacceptable".
After a successful interim spell in charge in the second half of last season, the former England midfielder was appointed on a permanent basis just five months ago.
But his position is at risk ahead of Sunday's visit of Bournemouth to Old Trafford.
United are off to their worst start in the Premier League era, were dumped out of the League Cup by Brighton and have lost two of their opening three Champions League games.
"We're in a moment where it's not gone well so we need to show exactly what it means. Respect this football club, respect what it takes to be here, respect what it takes to win, what you have to do to win," Carrick said in his pre-match press conference, which began with a six-minute impassioned speech.
"We've kind of lost our way a little bit if I'm totally honest and we'll get it back. I'm totally confident we'll get it back."
Fernandes decision swerved
Carrick's tactics were publicly questioned by Fernandes after the Como defeat for being too passive.
Despite being repeatedly probed on whether last season's Premier League player of the year will retain the armband, Carrick refused to clarify the Portuguese midfielder's future.
"The most important thing for me is the team, first and foremost, is the squad, is the dynamics in the squad," added Carrick.
"Again, it's about the football club. I always do what's best for the football club."
Fernandes's criticism came just over a week after defender Lisandro Martinez took aim at the club's board for a lack of investment in the transfer market.
Carrick is just the latest manager struggling to turn around the ailing fortunes of the 20-time English champions since his former boss Alex Ferguson retired in 2013.
United have not won either the Premier League or Champions League in the 13 years since.
Appointed on a short-term deal in January, Carrick led the Red Devils back into the Champions League for the first time in three years with a third-placed finish in the Premier League.
But their only two victories in 11 matches in all competitions this season have come against newly-promoted Ipswich and Azerbaijani side Sabah.
"Recently we haven't had a good time. I totally accept that. We had five months of relative success in a spell where I was happy with a lot of things we were doing as a team," Carrick said in his opening address.
"We haven't done well enough lately. We've had a spell we're really not happy with. Como was a bad night. It hurt me. It was really unacceptable.
"The supporters have to make the effort and spend the money to come and watch us. To have the feeling we all had after the game was a tough moment."
He added: "My job is to put a team on the pitch that represents the supporters and that they can be proud of the identity we show. We have gone away from that recently in certain games which I am fully aware of and definitely doing something about."