Bayern were reduced to 10 men after eight minutes against bogey side Gladbach, when France defender Dayot Upamecano was ruled to have brought down a charging Alassane Plea on the edge of the box.
Replays showed minimal contact, although the decision survived a subsequent video review.
"Emotions are a part of sport," Nagelsmann told reporters. "However I must apologise for my choice of words against the match official team of (referee) Tobias Welz.
"Unfortunately, I clearly went too far."
Journalists outside the room heard Nagelsmann say "that's a joke, they're taking the piss", leaving the room calling the officials "spineless".
Speaking with German television channel Sky after the match, Nagelsmann - who had approached the referees on the pitch - complained "nobody can explain to me (why) that was a red card".
The loss was only Bayern's second of the season and was the second time in three league matches they have been reduced to 10 men.
Bayern will drop to second if Union Berlin beat bottom club Schalke at home on Sunday.
