The announcement comes less than a week after Martyn Thomas' election as RFU management board chairman following his victory over Jack Rowell at a special general meeting of the RFU in London.
"It has been a privilege to work with the RFU in a variety of capacities over the last nine years," said Cotton.
"However, I have found it increasingly difficult to balance my business and RFU-related interests in the last year, and so regrettably I have had to make this decision."
RFU president Malcolm Phillips added: "We are sorry to lose one of the icons of the game.
"Fran has been a dedicated servant to rugby union through a successful playing career, his management of the 1997 Lions and as an administrator.
"Fran has acted as an RFU representative on the Six Nations, IRB and as a member of the RFU Council and management board. His work at all levels has been invaluable."