Serbia s Aleksandar Mitrovic controls the ball during the Euro 2024 group G qualifying soccer match between Serbia and Lithuania at the Rajko Mitic Stadium in Belgrade, Serbia, Friday, March 24, 2023. AP
Mitrovic was shown a straight red card for grabbing Kavanagh after the referee awarded a penalty and dismissed Fulham's Willian for deliberate handball on the line.
The Football Association have since announced the standard punishment for Mitrovic's dismissal - a three-game ban - would be "clearly insufficient".
Fulham were leading 1-0 at the time of the red cards, but went on to lose 3-1.
"On a personal level, I regret my actions that led to me being sent off. I allowed my frustration to get the better of me, and how I reacted was wrong," said Mitrovic in a statement posted on the club's website.
"I have accepted the three-match ban for my red card, I have spoken to Chris Kavanagh to apologise, and I have volunteered to accept a club fine.
"I now want to do what I can to put this incident behind me and get back to helping my teammates on the pitch as soon as possible."
Fulham boss Marco Silva was also shown a red card at Old Trafford for his protestations towards the officials.
Silva has been charged with multiple misdemeanours, including improper conduct, improper behaviour and abusing match officials.
"I regret what happened and I have spoken to Chris Kavanagh to apologise," said the Portuguese coach.
"He knows that I respect him and the job he has to do. Unfortunately, we did not see eye-to-eye on the day, but I recognise that he is one of the top referees in this country and I know that, when our paths cross again, there will be mutual respect between us."
There have been calls for Mitrovic to be banned for up to 10 games.
The loss of the 28-year-old could prove vital in Fulham's unexpected challenge to qualify for Europe.
Silva's men sit ninth in the Premier League, three points off seventh, which is likely to be enough for a place in the Europa Conference League next season.
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO_Sports and on Facebook at AhramOnlineSports.)