Depay scored the late equaliser that sent Lyon into the round of 16 but tempers flared after the players went towards the stands to celebrate with the fans.
After seeing a banner telling defender Marcelo to "go away," Depay chased the fan carrying it.
His teammates ran after him as other fans came down from the terracing.
"The management will have to take responsibility," Depay was quoted as saying in French media on Wednesday.
"You qualify and you leave the pitch in these conditions? Something is wrong. I chased the fan to tell him to drop the banner.
"They have spat on us, on members of the staff, some players -- I have never seen anything like this.
"I am upset and angry. I do not know what to say. We did not put in our best performance. However, we qualified. The people on the pitch gave everything. We knew how important this match was and they know how hard it can be.
"It makes no sense to play in a full stadium if we cannot celebrate with the fans afterwards. They do not want to be as one with their team."
Lyon are seventh place in Ligue 1, nine points short of Champions League qualification.