England and Russia were warned Sunday they could be disqualified from Euro 2016 if there is further violence involving their fans at the tournament.
UEFA's executive committee said both federations could be disqualified irrespective of any disciplinary measures taken if there is a repeat of the scenes in Marseille around Saturday's match between the two nations.
Disciplinary proceedings were earlier opened by European football's organizing body against the Russian Football Union after fans appeared to charge England supporters at the end of the match.
Rival supporters and locals had been involved in clashes in the days ahead of the game and further violence in the city took place before and after the match as police used tear gas for a third straight day.
In a statement UEFA's executive committee said it "would like to express its disgust for the violent clashes" in Marseille.
"Such unacceptable behaviour by so-called supporters of the national teams of England and Russia has no place in football, a sport we must protect and defend," it said.
The executive committee said that independent of any disciplinary sanctions relating to incidents inside the stadium "it will not hesitate to impose additional sanctions on the Football Association (FA) and the Russian Football Union (RFS), including the potential disqualification of their respective teams from the tournament, should such violence occur again."
Both associations are urged "to appeal to their supporters to behave in a responsible and respectful manner."
UEFA aims to rule by Tuesday on the disciplinary proceedings opened against the Russian association. It said it would be improving stadium security across the tournament after acknowledging "there were segregation issues at Stade Velodrome."
In the incidents following Russia's stoppage-time equalizer in the 1-1 draw, charging fans appeared to force England supporters to flee, with some having to climb fences and women and children seen in a state of panic.
In clashes in Marseille ahead of the game, supporters could be seen attacking each other with chairs and metal rods. One man was reported in a critical condition, with at least 34 others injured and 10 people arrested.
French Interior Minister Bernard Cazeneuve said France would now ban alcohol sale and consumption in certain high-risk areas.
The alcohol ban will be enforced the day before and the day of matches. Local authorities could also ban restaurants from having glasses or bottles on pavement terraces.
England FA chief executive Martin Glenn said: "We take this letter from UEFA with the utmost seriousness.
"We understand the potential implications of our supporters’ actions and wholly accept that every effort needs to be made by The FA to positively urge them to act in a responsible and respectful way."
Marseille police chief Laurent Nunes told France Info radio that "a quick reaction" and "the determination of the officers clearly led to preventing even worse incidents."
Local police has mobilized more than 1,000 officers to keep the fans in check.
Russian Sports Minister Vitaly Mutko was quoted in Russian media as saying security measures in the stadium were not up to scratch.
"You must organise such games well and separate the fans," Mutko said.
Mutko, who is also Russian football federation chief, said he expects a fine from UEFA and pledged to punish the offenders once they have been identified.
World football's governing body FIFA said in a statement to dpa that Russia, host of the 2018 World Cup, must learn "lessons" from the Euros in its planning for the big event.
French media reacted with shock, with sports paper L'Equipe speaking of "the shame" and Le Parisien of "scenes like in a civil war."
It said the violence was an insult to France, which has remained in a state of emergency ever since the November 13, 2015 terror attacks.
The violence has also spread to the city of Nice while there were clashes involving German fans ahead of Germany's match in Lille against Ukraine on Sunday.
Police in Nice said three people were arrested and nine injured in a fight during the night between fans from the local ultra scene and supporters from Poland and Northern Ireland, whose teams played on Sunday.
Two fans were slightly injured in Lille in a confrontation involving German and Ukraine supporters which was quickly ended by police, a local authority spokeswoman said.