Uefa had threatened to disqualify Russia after violence from fans.
"After a type of tournament like this, you need someone else to take charge of the national side for major competitions," Slutsky said.
The 45-year-old CSKA Moscow boss succeeded Fabio Capello as Russia coach last July, but only had a contract until the end of Euro 2016.
He was told by the Football Union of Russia he would have to choose between the national team and CSKA at the end of the tournament.
The 2018 World Cup will be held in Russia, who qualify automatically as hosts.
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