RFF has dropped an appeal against Fifa's World Cup suspension that was imposed after the country's invasion of Ukraine.
The Football Union of Russia had protested against the ban but the Court of Arbitration for Sport (CAS) says the FUR has now withdrawn that appeal
Russia had qualified to face Poland in a play-off semi-final but the Poles instead received a bye to face Sweden.
However, Russia will continue to fight bans from other sporting competitions.
Those include Uefa's ban on Russian sides competing in European club tournaments, such as Spartak Moscow's ejection from the Europa League.
Uefa have also prohibited the Russian women's national team from competing in Euro 2022 in England this summer as a number of countries have refused to compete against Russian opposition.
The 2022 Champions League final, originally due to be played in St Petersburg on 28 May, has also been moved to Paris.
Russia has also filed appeals seeking the annulment of its bans from international gymnastics, rugby, rowing, and skating events.