The game at Easter Road finished goalless but exploded into acrimony in the final few minutes when a bad tackle by Morton’s Kudus Oyenuga on Hibs defender Jordon Forster sparked a flare-up on the touchline.
Managers Neil Lennon and Jim Duffy had to be pulled apart and McGregor was shown the red card for confronting Oyenuga, who was also sent off.
Lennon, who was sent to the stand with Duffy, branded Oyenuga’s actions “disgraceful”, accusing him of trying to break Forster’s leg then pretending he had been headbutted by McGregor.
“We will deal with it accordingly and we will appeal Darren’s red card because that was embarrassing,” Lennon said of Oyenuga’s actions.
“The boy feigns an injury as if he’s been headbutted really badly. It’s disgraceful. No wonder I was angry.”
Hibs are due to play Dunfermline away on Saturday and then face Morton again on 8 April, at Cappielow.