The Brazilian landed himself in hot water last month after a video posted by a spectator at the Coupe de France final appeared to show him striking a fan after a brief confrontation.
The French Football Federation (FFF) announced the sanction on Friday, with the punishment also including a two-game suspended ban.
Unhappy with what it considers the severity of the sanction, PSG is contesting the ban with the FFF High Commission of Appeal.
"Given the insults endured by several Paris players, including Neymar, at the end of the Coupe de France final, and the various elements submitted to the committee by Paris Saint-Germain in the player's defence, the club considers this sanction severe," said a PSG statement.
Neymar met une pêche à un rennais pic.twitter.com/uXgU3R2l5Y— Claude vaillant (@vaillant92100) April 27, 2019