The Barclays Premier League champions were six minutes away from a memorable victory at the Bernabeu before a late turnaround saw them crash 3-2 in their Champions League opener.
Former Manchester United star Cristiano Ronaldo grabbed the winner in stoppage time after having an effort beaten away by the inspired Hart - one of several fine saves by the keeper - moments earlier.
Hart told ITV: "You can't go 2-1 up with five minutes to go and lose the game. Who else can we blame? We can only blame ourselves.
"We dug deep, got a lead twice and we lost it, so we can only blame ourselves.
"It's hard to come off the field after losing 3-2 and be positive. I'm really sorry.
"We're not a team that should come here and pat ourselves on the back for doing well. We're a team that gets results."
While Hart's comments might have been an honest appraisal of the match, manager Roberto Mancini felt he should keep such remarks to himself.
Speaking in his post-match press conference, Mancini said: "Joe Hart should stay as goalie and finish.
"If anyone should criticise the team, I should criticise the team, not Joe Hart."
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City were second best for most of a compelling Group D clash but remained on level terms - mostly thanks to Hart - until Edin Dzeko fired them into the lead against the run of play after 68 minutes.
Marcelo converted one of a number of chances to level but City looked like snatching victory when Xabi Alonso's outstretched boot was unable to prevent Aleksandar Kolarov's 84th-minute free-kick from out wide bouncing in.
But the hosts produced a rousing finale and Karim Benzema equalised from the edge of the area before Ronaldo had the final say.
Mancini said: "It is difficult not because we lost - we can lose against a side like Real Madrid - but because with four minutes to go it was 2-1.
"This is strange. I think we need to improve this situation.
"Sometimes we go too deep. We made some mistakes and I don't want us to continue defending like this.
"In the first half they played better than us but in the second half we played well and scored two goals.
"When you score two goals here and are 2-1 you should pay more attention."
City are only in their second Champions League campaign and failed to make it past the group stage last season.
Further tough games against Ajax and Borussia Dortmund lie ahead but Mancini will not use inexperience as an excuse.
He said: "We can't continue to talk about experience. We have good players who play for their international teams. They play a lot of games.
"I think I am happy for our performance in the second half but I think we can improve and we should be disappointed that we lost."