The disciplinary proceedings opened against the Football Union of Russia are regarding crowd disturbances, racist behaviour and fireworks being set off at the Stade Velodrome in Marseille.
The European governing body has not taken any action against the English Football Association.
A large group of Russian fans in a stand behind England's goal advanced on the neighboring Three Lions section at the full-time whistle, throwing objects and breaking through a line of stewards, who were outnumbered and offered little resistance.
England fans quickly fled for the exits as Russia fans gave chase and snatched English flags that had been on display.
In addition to the confrontations between the rival sets of fans, fireworks were set off inside the stadium, raising questions about security; the fireworks should not have been allowed inside the ground.
Two large white flares were set off in the Russian end of the ground, while fireworks were also let off, with a large bang heard from behind the goal defended by England's Joe Hart.
Shortly before the final whistle, Russia fans also appeared to seize a power cable running around the front of the stand.
UEFA said in a statement: "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 Velodrome. This kind of behaviour is totally unacceptable and has no place in football.
"Disciplinary proceedings have been opened against the Russian Football Union (RFU) for the following events occurring inside the stadium: crowd disturbances, setting off of fireworks and throwing of missiles.
"A decision on the sanctions to be imposed will be made within the next few days, once the RFU has been able to submit written statements and evidence.
"UEFA acknowledges that there were segregation issues at Stade Velodrome and will implement corrective measures to strengthen the deployment of security personnel at stadiums, in close collaboration with local authorities.''
Sanctions will be decided at a UEFA disciplinary meeting on Tuesday. The disciplinary panel could take into account Russian fans' track record of violence, including at Euro 2012, when stadium security staff were attacked in Poland.
A fine from UEFA is the most likely punishment to be handed out, although Russia could face a possible Euro 2016 points deduction.
"We will have a fine from UEFA, so I understand. We behaved incorrectly," Russia's Sports Minister Vitaly Mutko is quoted as saying by R-Sport news agency.
FIFA released a statement condemning the violence seen around matches at Euro 2016, with fighting also taking place in Nice ahead of Sunday's game between Poland and Northern Ireland.
"FIFA strongly condemns any form of violence and finds it wholly unacceptable to see such shameful scenes around football matches perpetrated by a minority of idiotic troublemakers, who have nothing to do with football and its true fans," the statement read.
"Events such as UEFA Euro 2016 are attended by millions of people from around the world simply with the intention of celebrating football and enjoying the matches, and that is how it should always be."
The trouble at the Stade Velodrome came after a third day of violence in the Mediterranean port city left as many as 20 England fans injured, with reportedly several seriously hurt in bloody clashes between rival fans.
Fist fights and bottle throwing broke out between the England supporters and their Russian and French counterparts on Saturday afternoon, with French police using water cannons and tear gas on rioters hours before the game started.
Rebekah Vardy, the wife of England striker Jamie Vardy, wrote on Twitter that she was subjected to tear gas before kick off, though it was unclear whether it occurred within the stadium.
That has to be up there with the worst experience EVER at an away game! Teargassed for no reason, caged and treated like animals! Shocking!
- Rebekah Vardy (@RebekahVardy) June 11, 2016
"That has to be up there with the worst experience EVER at an away game! Teargassed for no reason, caged and treated like animals! Shocking!," she wrote. "I witnessed this with my own eyes! I can't comment on things I didn't see but what I got caught up in was horrific and uncalled for!"