England are facing the threat of disqualification from Euro 2016 if the widespread violence and running battles that caused chaos across the streets of Marseille are repeated.
Uefa branded the behaviour of England and Russian fans before and after their 1-1 draw at the Stade Velodrome as "unacceptable" and said it would not hesitate to impose additional sanctions.
It also opened disciplinary proceedings against the Russian Football Union for alleged crowd disturbances, racist behaviour and the setting off of fireworks by its fans during the game on Saturday night.
It appealed to both country's football bodies to urge supporters to "behave in a responsible and respectful manner", adding: "The Uefa executive committee has warned both football associations that - irrespective of any decisions taken by the independent disciplinary bodies relating to incidents inside the stadium - it will not hesitate to impose additional sanctions on the Football Association (FA) and the Russian Football Union (RFU), including the potential disqualification of their respective teams from the tournament, should such violence occur again."
A Downing Street spokeswoman said six British nationals remain in hospital after supporters fought with Russian and local French hooligans over three consecutive days.
The spokeswoman said, while UK officers have been in France since before the tournament began, they had offered to send more "to support the security operation around the match in Lens".
She added UK police would be assisting the French with their post-incident investigations and that Home Secretary Theresa May had spoken to her French counterpart, Bernard Cazeneuve, the interior minister, to continue to co-ordinate and work closely together on the response.
Russian sports minister Vitaly Mutko has also condemned the violence.
England fans were seen fleeing the stands shortly after the final whistle was blown as balaclava-clad thugs punched through a line of stewards before attacking bystanders.
The incident raised serious questions about French security at a time when the country is on high alert after November's terrorist attacks on Paris.
The scenes from Marseille were described by the FA as "terrible" as it called on the local authorities to deal with the perpetrators swiftly.
England fan Sam Blackwell was at the Stade Velodrome on Saturday night and told the BBC that security had been "very poor" with stewards left to cope on their own for five to 10 minutes before help arrived.
He said: "Even before we got in the stadium we were surprised how slack the searches were going into the ground, because they were almost non-existent. It was just a quick pat down and then they let you through.
"If they were doing them properly how would the Russian fans have got the fireworks and the flares in?"
He added: "I would say it was very poor security given all I have been reading in the press about how it was going to be at its highest because of the risks that there were from other threats. I thought it was very poor."
Struggling to make the long journey from Marseille to Lens for England's next fixture against Wales due to strikes affecting planes and trains, supporters were also hit with the news that the French government had banned the sale of alcohol from "sensitive areas" on match days.
Making the announcement, Mr Cazeneuve condemned the events in Marseille as unacceptable and said the police enforcement system which was in place had been "properly-sized" and had "helped restore calm in an hour and a half, separating the highly alcoholic protagonists".
Culture, Media and Sport Secretary John Whittingdale, who said he welcomed the Uefa investigation.
He said: "The scenes that occurred between a small minority of fans in Marseille were extremely concerning. The vast majority of supporters in France simply want to play their part in a great festival of football.
"We want fans of all teams to enjoy themselves but to be respectful of others and the local authorities. We urge supporters to remain calm.
England fans returning from France on the Eurostar at St Pancras on Sunday were met by about a dozen police officers appealing for information about the trouble in Marseille.
The officers were asking for witnesses and handing out leaflets to passengers.