A French prosecutor says there were around 150 Russian "hooligans" who travelled to Marseille "well-prepared for violence"
The French ambassador to Moscow has been summoned to the Russian foreign ministry after sharp criticism of policing at the Euro 2016 tournament.
"Further stoking of anti-Russian sentiments... could significantly aggravate the atmosphere in Russian-French relations," the ministry said.
Foreign Minister Sergei Lavrov rebuked French police for detaining 43 Russian fans after clashes in Marseille.
Separately, France is to expel four Russians arrested in Lille.
France's crackdown on hooliganism among supporters relates to incidents outside the stadiums.
Uefa, football's European governing body, separately fined Russia and gave it a suspended disqualification following fan violence inside the stadium in Marseille where Russia played England on Saturday.
The Russian foreign ministry summoned Ambassador Jean-Maurice Ripert soon after a speech by Mr Lavrov to the lower house of the Russian parliament in Moscow.
Mr Lavrov suggested Russian fans had been provoked and criticised the way French police were subjecting them to security checks.
"It was an absolutely unacceptable incident when a bus with more than 40 Russian supporters was stopped and [the police] demanded that they leave the bus for document and ID checks," he told the State Duma.
He accused the French of violating international conventions by detaining the fans.
"It is a fact that the French behaved completely contrary to their obligations under the Vienna Convention, and I have already written to the French foreign minister, demanding that he does not allow any more such incidents to occur."
Mr Lavrov did concede that the behaviour of some Russian fans at the tournament had been poor.
"Behaving like some of our citizens did, bringing flares, fireworks and so on, is unacceptable."
"However..." he added, "we cannot close our eyes to the attempts to ignore the provocative actions of other countries' fans".
French police blamed 150 "well-trained" Russian hooligans for clashes before England played Russia.
French police stop a coach carrying Russia fans
The bus carrying 43 fans was stopped in a police operation near Cannes on Tuesday, and some of those Russia supporters are now being deported.
Among them was far-right sympathiser Alexander Shprygin, who heads a Russian football supporters' association.
Five England fans were jailed for throwing bottles at police and a sixth jailed in connection with the violence.
A match between Russia and Slovakia in Lille on Wednesday ended with the Slovaks winning 2-1.
There was no significant crowd trouble inside the stadium, although a flare was ignited by Russia fans just after their team had scored.
Lille is also being used as a base by supporters of England and Wales, who are due to play in nearby Lens on Thursday.
There are reports of tear gas being fired in Lille on Wednesday afternoon after brief clashes between fans and police, but it is unclear which fans were involved.
The authorities in Lille announced that four Russians would be expelled, two for a scuffle near a railway station, and two who had been found with a baseball bat and a hammer in their car during a vehicle check.
Three other Russians were detained on Wednesday morning, it added.
The French authorities have pledged to increase security in Lens, deploying 2,400 police, gendarmes, security guards and a riot squad.
Drinking alcohol in the streets has also been banned.
Up to 50,000 England fans are expected in Lens despite the fact that all 35,000 tickets had sold out.