Muller was given his marching orders in the 75th minute after accidentally scraping his studs on Tagliafico's head when trying to control a high cross-field pass.
The Argentina international suffered a cut but no serious lasting damage and was able to complete the match – later scoring a last-gasp equaliser.
The 29-year-old may be dealt more than the manadatory one-match ban if UEFA deems the incident a serious offence at a hearing on January 10.
Muller's dismissal was the catalyst for a thrilling conclusion to the contest, as Dusan Tadic, Robert Lewandowski, Kingsley Coman and Tagliafico all found the net from the 82nd minute onwards.
But it was the red card which spawned the most debate and Muller was apologetic on Thursday.
"I would like to apologise to Nico Tagliafico for yesterday's incident," Muller wrote on Twitter. "There was no intention [in the foul]. Get well."
Auf diesem Wege möchte ich mich bei Nico Tagliafico für den gestrigen Vorfall entschuldigen. Das war keine Absicht. Gute Besserung pic.twitter.com/TsMcOxdlpi— Thomas Müller (@esmuellert_) December 13, 2018
Bayern boss Niko Kovac acknowledged after a dramatic contest in Amsterdam that his World Cup-winning forward had deserved to be dismissed, even if his actions were accidental.
The Croat said: "The red card for [Maximilian] Wober was totally justified because he took on that challenge in a negligent way.
"The red card for Thomas was fair as well. He didn't see the player but you have to be aware of your opponent coming at the ball at your back. Both penalty decisions correct. Both red cards correct.
"The referee did well, except in the case of the goal that made it 3-3. I'm not a referee but I think it was offside."
Bayern and Ajax both qualified for the knockout phase on the back of a thrilling stalemate, with the Bundesliga giants finishing top of Group E.