The incident took place Tuesday during a 2-1 friendly loss to home side Sassuolo in the Trofeo TIM tournament, which also involved Serie A champion Juventus.
Constant, who is black, walked off the field after kicking the ball into the stands in the 34th minute, emulating the actions of Kevin-Prince Boateng last season. The referee allowed a substitute to come on and also ordered an announcement warning that the game would be suspended if further incidents occurred.
"Racist chants are despicable and scandalous and they should be fought against and I want to stress that," Milan vice-president Adriano Galliani said. "But the rules are that when they happen, you should turn to the referee, who will tell the fourth official, who will advise the person responsible in the police, who is the only one authorized to suspend the game.
"There is solidarity with Constant, but it is not OK to leave the pitch. I have told him, repeated it and written it to everyone: You can't leave the pitch."
Following racist abuse during a friendly in January against Pro Patria, Boateng also kicked the ball at a section of the crowd before walking off the field with the rest of the Milan team.
FIFA said it is "monitoring the situation" and awaiting the outcome of the Italian football federation's investigation.
"FIFA's position on the issue of racism is unequivocally clear: There is no place in football for racism or for any form of discrimination," FIFA said in a statement.
Shortly after the incident, Constant tweeted a mock-up movie poster of teammates Boateng, Mario Balotelli and M'Baye Niang and the title "Stop Racism."
Sassuolo is about to start its first ever season in Serie A after winning the second division last season.