"Further stoking of anti-Russian sentiments... could significantly aggravate the atmosphere in Russian-French relations," the ministry said.
Russia`s foreign minister singled out the detention of 43 fans on a bus. France`s embassy insisted that the arrest of Russia fans was "absolutely according to the law".
Meanwhile, four Russians arrested in the French city of Lille are to be deported but a match there between Russia and Slovakia on Wednesday passed off peacefully.
France`s crackdown on hooliganism among supporters relates to incidents outside the stadiums.
Uefa, football`s European governing body, earlier 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 Foreign Minister Sergei Lavrov to the lower house of the Russian parliament in Moscow.
Mr Lavrov suggested Russian fans had been provoked and criticized the way French police were subjecting them to security checks.
Russia summons French ambassador after fan violence
16 June 2016
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The French ambassador to Moscow has been summoned to the Russian foreign ministry after sharp criticism of policing at the Euro 2016 tournament.
