His decision was announced on the same day that Williams declared he would be appealing against the severity of the one-year ban imposed on him for his part in the affair.
Harlequins were fined a world record 250,000 Euros by an independent disciplinary panel after it was established that Williams' left the field against Leinster with a fake blood injury, but misconduct charges against Richards, club doctor Dr Wendy Chapman and physio Steph Brennan were dismissed at the time.
However, tournament organisers ERC have now appealed against the brevity of the sentencing against the club, Richards and the two medics.
Richards resigned from his position as director of rugby at the Guinness Premiership side after the club instigated an internal investigation into the affair
In a statement issued by the club they confirmed that "Dean Richards unreservedly tendered his resignation from his post as director of rugby and the board of Harlequins has accepted his resignation."
Williams was spotted winking towards the bench as he left the field shortly after coming on as a replacement with blood pouring from a fake injury in his mouth. His departure allowed Nick Evans to return, in the hope the All Black outside half might slot a match winning drop goal in the final five minutes of the 6-5 defeat against the eventual champions.
The Harlequins statement added: "Following the hearing held and decision made by the ERC panel on 20 July, the club has conducted its own review of the incident. This review is still ongoing.
"As part of this process, the club has waited for the written judgment to ensure it had considered carefully the concerns and position of the ERC disciplinary panel.
"The club fully accepts the verdicts and sanctions imposed upon the club since it is accountable for the failings of its staff. The club feels strongly that the severity of the sanction meted out to Tom Williams is unfair and disproportionate and the club will support him in the event of an appeal on his part against the 12-month playing ban currently in place.
"We acknowledge that this whole episode has tarnished the image of the club, which has always sought to uphold the highest standards. The level of scrutiny that clubs such as Harlequins now find themselves under means that procedures and processes which may have served us well in the past are no longer adequate.
"The club is urgently undertaking a thorough review of its internal policies with a view to establishing an appropriate ethics code for all staff (both playing and non playing) covering a wide range of areas including respect for the laws of the game."