Marseille's Payet was struck in the back by a bottle thrown from the home supporters as he went to take a corner.
He launched it back, leading to fans pouring on to the pitch as stewards and players struggled to restrain them.
After a long delay, Nice's players came out to finish the game - but Marseille's players refused to.
On Monday, Ligue de Football Professionnel, the French league's governing body, said that following the "serious incidents", both clubs had been summoned to its disciplinary commission on Wednesday.
French Sports Minister Roxana Maracineanu has called what happened "an insult for sport and football".
While the mayor of Nice Christian Estros said: "This violence is unacceptable. Sanctions must be taken by the French football league once they have found out who was responsible."
"Our players were attacked," said Marseille president Pablo Longoria.
"We decided for the safety of our players, who were attacked during the pitch invasion, not to resume the match because the safety of our players was not guaranteed."
The chaos continued as the players tried to leave the pitch, with scuffles breaking out and Marseille coach Jorge Sampaoli having to be held back by his staff.
Marseille players had been targeted by plastic water bottles from that section of the crowd throughout the match, before one hit Payet and saw the midfielder fall to the ground.
A security cordon of stewards tried to stem the pitch invasion before a brawl broke out between players of the two teams, supporters and staff.
Players and officials eventually made it off the pitch and back down the tunnel away from the chaos outside.
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