The Court of Arbitration for Sport (CAS) said on Tuesday the Football Union of Russia (FUR) had withdrawn its appeal against football's world governing body FIFA as well as the Polish, Swedish and Czech Republic Football Associations.
FIFA and European governing body UEFA decided together that all Russian teams, whether national or club sides, would be suspended from participation in their competitions until further notice after Russia's invasion of Ukraine.
Russia were set to host Poland in a World Cup qualifying playoff on March 24 but FIFA had said that Poland would be given a bye to the final of their path in the playoffs to play either Sweden or the Czech Republic.
CAS said that the procedure would be terminated shortly.
Russia, which calls its actions in Ukraine a "special military operation", has also filed appeals seeking annulment of its bans from international gymnastics, rugby, rowing, and skating, CAS said.
Last month, Russia's hopes of competing at the 2022 World Cup in Qatar were effectively ended after CAS rejected a request from the FUR to freeze FIFA's suspension while their appeal was being heard.
Several national football associations announced they would not play against Russia in the wake of its invasion of Ukraine, including those of Poland, Sweden and the Czech Republic.
Poland went on to beat Sweden in their playoff match to qualify for the World Cup finals, which begin in November.
CAS had also upheld UEFA's ban on Russian teams taking part in its competitions for the duration of its proceedings.