Sections of both sets of fans have been involved in violent clashes since the tournament kick off on Friday
UEFA has warned England and Russia they could be disqualified from Euro 2016 if there is further violence between supporters of the two nations.
It comes after Russia was charged with crowd disorder and racist behaviour after ugly scenes overshadowed Saturday's game at the Stade Velodrome, which ended 1-1.
England supporters were forced to flee the stands shortly after the end of the match as Russian spporters broke through a line of stewards and started attacking them.
The disorder was the climax to three days of violence between English, Russian and French fans in the port city that left dozens injured and led to more than a dozen arrests.
"Uefa expresses its utter disgust for the violent clashes that occurred in the city centre of Marseille, and its serious concern for the incidents at the end of the match inside Stade Vélodrome," it said in a statement.
"This kind of behaviour is totally unacceptable and has no place in football."
Tempers flare between Russia and England fans during the UEFA Euro 2016, Group B match at the Stade Velodrome, Marseille. Image: Nick Potts / PA Wire/Press Association Images
The British Government has said it was "deeply concerned" by the violence, and has offered to send more police to France.
"The UK government is deeply concerned by the violence in Marseille last night, including reports that in some cases England fans were attacked by rival supporters," a spokesman said in a statement issued by Downing Street.
"We have offered to send further UK police to France ahead of the next England game to support the security operation around the match in Lens.
"And UK police will be assisting the French with their post-incident investigations and supporting them to gather evidence, including evidence against any England fans involved in the disorder."
Russia has also been charged over fireworks being set off inside the ground.
Vitaly Mutko, the Russian sports minister who led its successful bid to host the 2018 World Cup and sits on Fifa’s executive committee and insisted in the wake of the trouble in Marseille that it had been exaggerated and blamed organisers for poor segregation.
"There was no clash ... That’s being exaggerated, in fact everything is fine here," Mutko told R-Sport.
"When the match ended, there was no barrier between the fans. The British were upset, of course, but it all quickly dissolved. Such matches should be organised properly. It is necessary to separate the fans [at the stadium].”
Flares and a firecracker - both of which were banned from the stadium - were set off at the end of the game, before Russian fans appeared to attack English supporters.
UEFA says its control, ethics and disciplinary body will decide what action to take against Russia on Tuesday, before the team's second game against Slovakia the following day.
European football's governing body has not brought any charges against the Football Association in the aftermath of the violence.
Russia's sports minister Vitaly Mutko has criticised the behaviour of his country's fans following the clashes.
He admitted UEFA was likely to impose a fine - but also laid the blame on match organisers for failing to separate supporters.
The Russian Football Union said it "expresses regret for the disturbances involving Russian fans both outside and actually at the game," and called on fans to "respect the opponents and their fans."
UEFA has acknowledged there were segregation issues at the Stade Velodrome, and said it will act to improve security at stadiums.