UEFA said on Saturday it was investigating the incidents which took place during the 4-1 win over the Czech Republic in the city of Wroclaw on Friday.
"Some people in the stadium carried out acts which were unworthy of real soccer fans. There is no place in the stands for people who use sporting events to declare personal, political or other points of view," the federation said in a statement on its website.
"The federation and the Russian national team strongly ask all real fans to oppose the provocative acts of hooligans and to fully cooperate with match organisers when it comes to questions of security."
European soccer's governing body is probing reports of alleged abuse directed at Czech players by Russia fans.
"We are appealing to all fans. Remember, you represent your country. Respect yourself, your motherland and your team," said the federation statement.
UEFA is also investigating what sparked a fight after the Wroclaw match in which about 30 fans attacked stewards.
The head of the Polish company in charge of tournament coordination said four stewards were hurt after being set on by a group of drunk Russia fans.
In a separate incident, two Russia fans aged 27 and 41 appeared before a court in Wroclaw and were issued with a ban on taking part in mass events. They also had their passports confiscated until they pay a 2,000 zloty ($580) fine.
Wroclaw police spokesman Kamil Rynkiewicz told Reuters the two were detained before the start of Russia's match because "they refused to follow the orders of peace-keeping services".
They spent a night at the so-called "sobering-up tank".