The Exiles were docked five points for fielding an ineligible player earlier in the season, while their relegation to the second tier was confirmed following defeat to Northampton Saints on Sunday.
Copsey told the club's official website: "I'm obviously very disappointed to be stepping down as the chief executive of London Welsh, but given the situation I did not feel carrying on in my current capacity was the best for me or the club.
"I will continue to support the club going forward and wish it all the best in the very challenging times it faces ahead.
"I know it has the players, the staff and the spirit to overcome the difficulties it has had to contend with in recent weeks and would like to thank all the staff, players and supporters, who have been tremendous in their support and professionalism through increasingly difficult times."
The club have lost their last 10 league matches in a row.