Grosso sustained face injuries in another episode of fans brutality to rock Ligue 1, which two years ago was marred by several acts of violence and crowd troubles.
Since the incidents happened outside Marseille's Stade Velodrome, the French League (LFP) have no jurisdiction to sanction the home team, although it could decide that the game has to be played behind closed doors.
The indicents are a bad look for the LFP, who have been negociating a TV rights deal for 2024-29, with the tender ending on Thursday, and made for embarrassing footage less than a year before the Paris 2024 Olympics.
Interior Minister Gerald Darmanin said nine people had been detained so far. "I hope the highest possible prison sentences will be handed to those fans that ruined the party for everyone," he added on BFM TV.
Some preliminary round Olympic matches will be played in Marseille, where violence erupted at the 1998 World Cup between England fans and French police.
"OL regrets that this type of situation recurs every year in Marseille and calls on the authorities to assess the seriousness and repetition of this type of incident before an even more serious tragedy occurs," Lyon said in a statement, with the club adding they would lodge a formal legal complaint.
"What happened to Fabio Grosso is totally unacceptable," OM presient Pablo Longoria said.
"This is something that cannot happen in football. Even if it happened outside of the stadium, it is unacceptbale. I am angry and appalled by the situation."
Marseille condemned the behaviour of their fans in a statement on Monday.
"Olympique de Marseille deplores the unacceptable incidents that took place in the vicinity of the Stade Orange Velodrome, involving the professional team bus and buses of Olympique Lyonnais supporters," the statement read.
"The club wishes Lyon coach Fabio Grosso a speedy recovery and strongly condemns this violent behaviour, which has no place in the world of football or in society."
Sports minister Amelie Oudea-Castera said: "These unacceptable acts deny the very values of football and sport, and the perpetrators must all be found and severely punished.
NAZI SALUTES
"We have seen just about everything you would not want to see," Sports Minister Amelie Oudea-Castera told France 2 TV, adding that fans had also been chanting homophobic and racist slurs inside Marseille's Stade Velodrome.
"It was distressing, revolting, disgusting."
Lyon condemned their fans' behaviour after some were seen making Nazi salutes at the Velodrome.
"OL firmly condemns the unacceptable racist behaviour of individuals in the stands on Sunday," the club said in a statement.
"The club has requested the footage in order to identify the perpetrators of any act contrary to the law, but also contrary to its values, and reiterates its determination to keep them away from the stands."
Seven suspects had been arrested on Sunday night, the minister added.
Sunday was not the first time that the away team bus has been targeted in Marseille, with previous incidents reported in 2016 and 2019.
It also came after fan trouble marred the 2021-2022 season.
That time Nice were docked two points and one of the players/fans? suspended, after serious incidents during their game against Marseille when visiting players clashed with home fans who pelted them with missiles and stormed on to the pitch.
The northern derby between RC Lens and Lille was also hit by crowd problems, with the start of the second half delayed after rival fans threw objects at each other before people ran on to the pitch, prompting riot police and stewards to intervene.
In that 2021-2022 season, there had also been crowd trouble in Montpellier, Angers, Marseille and St Etienne.
