The Argentina international saw red five minutes before half-time for an elbow into the chest of Armando Izzo as the two jostled for possession in midfield.
The 10-man Bianconeri were condemned to defeat by a 74th-minute goal from Monza substitute Christian Gytkjaer to further ramp up the pressure on head coach Massimiliano Allegri.
However, the coach was not to blame, according to Di Maria, who said the result was "all my fault".
In an Instagram post, the former Paris Saint-Germain winger said: "I want to apologise to everyone for this inappropriate reaction I had on the pitch.
"Leaving the team with one down at such a difficult time has cost us the game.
"Losing is all my fault. I am so sorry. I'm a professional but also a human being who makes mistakes and knows how to admit them."