Masked Russia supporters charged at England fans, punching and kicking them, after the final whistle of the 1-1 Group B draw at the Stade Velodrome in Marseille on Saturday.
Russia's fans were also involved in violent clashes with England supporters in the port city before and after the match.
"Charges relating to crowd disturbances, use of fireworks and racist behaviour had been brought against the RFU (Russian Football Union)," UEFA said in a statement, adding its ethics committee had imposed a suspended disqualification and fine.
"Such suspension will be lifted if incidents of a similar nature (crowd disturbances) happen inside the stadium at any of the remaining matches of the Russian team during the tournament," it added.
Sports minister Vitaly Mutko told the R-Sport news agency that Russia, who have been chosen to stage the 2018 World Cup, would not appeal against the sanction.
"This decision (to fine Russia) was pre-determined. There was the decision of the executive committee, now they just confirmed this decision," said Mutko.
"The punishment is excessive, but we cannot influence it. The fine is enormous as the Russian Football Union is a non-commercial organisation. There is no sense to appeal. But what (has) the team has to do with that? It is not guilty."
STRONG RESPONSE
The violence in the stadium followed three days of clashes between English, Russian and French fans in the Mediterranean port city that drew a strong response from riot police.
French Prime Minister Manuel Valls said on Tuesday that some football fans will be deported from France after violent clashes marred the start of the Euro 2016 tournament, and a group of Russia fans was detained near Cannes on the French Riviera.
While UEFA's sanctions announced on Tuesday relate to incidents inside stadiums, the governing body warned last Sunday that it could throw England or Russia out of the tournament "irrespective of any decisions taken by the independent disciplinary bodies relating to incidents inside the stadium" if violence continues in areas outside.
Russia next face Slovakia in Group B in Lille, while England take on Wales in neighbouring Lens on Thursday. Wales are top with three points followed by England and Russia with a point apiece and Slovakia who are yet to get off the mark.
Security has been beefed up for the two games, UEFA said.
"We will implement additional corrective measures for the games coming up in Lens and Lille and we hope that those measures will be sufficient to avoid any other crowd trouble during the matches," a spokesman said.
It is the second time in as many European Championship tournaments that the Russian federation has faced sanctions because of the behaviour of its supporters.
The RFU was fined and given a suspended six-point deduction for its Euro 2016 qualifying campaign after fans assaulted stadium security staff and displayed illicit banners at the 2012 tournament in Poland and Ukraine.
($1 = 0.8913 euros)
(Additional reporting by Angelica Ramos; Editing by Ken Ferris)
