Mexes was sent off late in his side's 3-1 loss at Lazio, having clashed with Stefano Mauri and grabbed the Lazio captain's throat.
The Frenchman also appeared to throttle Lorik Cana, and his team-mates had to restrain him before he left the pitch.
Afterwards, Mexes said sorry and expressed regret for the incidents.
"I have to apologise to the club, the coach and my family," the 32-year-old said.
"It was the wrong reaction and I am sorry because I left the team in 10 men and I will be suspended.
"Unfortunately I made a mistake, in my career I have had these reactions. These things should not happen out on a football pitch."
Milan had gone ahead through Jeremy Menez before conceding three second-half goals.
Mexes said the players were behind under-fire coach Filippo Inzaghi, whose side are winless in Serie A this year and sit ninth.
"We played well in the first half, even if we were slightly at risk. We were compact, it was all perfect and then they scored," Mexes said.
"We play them again [in the Coppa Italia] in three days and have to bounce back.
"We’re all close and with the coach. We will try and resolve this situation, which isn't nice, and to do so we have to win and not have reactions like mine tonight.
"Every match has to be like a final and the one on Tuesday even more so as we have to get back on track."
