McCoy, who suffered a season-ending knee injury in Week 1, said in a statement posted to Twitter that he was prescribed a banned substance "to help with scar tissue and tendon strength from a previous injury" and that it was "an honest mistake."
The NFL said McCoy will immediately begin serving the suspension without pay while on the team's reserve/injured list.
"I love the game of football and have nothing but respect for the players, fans, and this league," McCoy said in his statement. "I've given my all to this game and worked my entire career to compete, train, and rehabilitate at a high level with integrity.
"It is with great disappointment that I recently learned I tested positive for a banned substance. ... In no way would I ever intentionally take anything to help with performance or gain a competitive advantage. This was an honest mistake, but it's something I take full responsibility for. I apologize to my family, the NFL, my teammates, and the fans and ask humbly for your forgiveness."
The six-time Pro Bowler, who was placed on injured reserve, had his season end early because of injury for the second consecutive year. The 33-year-old suffered a ruptured right quadriceps in training camp with the Dallas Cowboys in 2020 before signing a one-year, $1.25 million free-agent deal with Las Vegas on Aug. 4.
ESPN's Paul Gutierrez contributed to this report.