The union headed by Sports Minister Vitaly Mutko expressed "regret over the disturbances with Russian fans participating both outside the stadium and at the match" against England in Marseille, France on Saturday.
Addressing fans, the RFU told them to "show respect to the opponent and its fans. Follow the laws and rules of conduct and public order in France."
Russia's matches are "under the intense scrutiny of international football disciplinary bodies," it warned.
UEFA made a public warning Sunday that Russia and England face disqualification if there is further fan violence. It called on both countries to appeal to their fans to behave.