Sutton was suspended Wednesday morning after the organization opened a formal investigation into alleged derogatory comments he had made.
Sutton had been due to oversee Great Britain's campaign at the Rio Olympics, which start on August 5, but he said that with just 100 days remaining until the event it was important the athletes could prepare without any complications.
"The developments over the past few days have clearly become a distraction," Sutton said. "It is for this reason, and having spoken to friends and family, that I believe it is in the best interests of British Cycling for me to step down from my position as technical director."
"Cycling is my passion and I have always worked to get the very best out of professional athletes," he added. "I am proud of what British Cycling has achieved and I am excited by the potential of the team for Rio. They will always have my full support."
British Cycling chief executive Ian Drake, who thanked Sutton for his work and the role he has played in the organization's success, confirmed that programs director Andy Harrison will take over with immediate effect.
"I understand and respect Shane's decision to stand down," Drake added. "His primary focus has always been the athletes, and this decision is something he has taken to allow them to focus on their preparation for Rio."
Discrimination allegations
Darren Kenny, one of Britain's most successful Paralympic cyclists, had told the Daily Mail that he had heard Sutton using discriminatory terms when referring to the British disability team.
"The term used to refer to us was generally 'gimps,' with another word in front of that," Kenny said. "The attitude towards them was abysmal. We were tolerated at best.
"I know others had an issue as well with not being allowed on the track and not being given time to prepare for competitions," he added.
Kenny's claims followed on from those of Varnish, who said she was subjected to the "go and have a baby remark" in a conversation with Sutton and head coach Iain Dyer after missing out on a place in the 2016 Olympics.
Last week British Cycling said it would be looking into the comments made by Varnish, and the organization confirmed on Tuesday it will put into place an independent review, in line with UK Sport, which will look at its "performance programs" following the fresh allegations of discrimination.
Drake reiterated that stance following Sutton's resignation, adding that the organization will "continue to be committed to promoting equality of opportunity and providing a supportive environment within British Cycling."
Sutton "wholeheartedly" denied Varnish's claims and said he only acted with complete professionalism towards her.
He was not immediately available for comment after a request from CNN following the latest allegations.
The 58-year-old Australian has been with British Cycling since 2002, when he was first appointed as a coach, and has worked with the likes of Olympic champions Chris Hoy and Bradley Wiggins.
He helped the team to seven track gold medals at the 2008 and 2012 Olympics, before being made technical director in 2014.
Varnish says she remains keen to ride for Team GB again.
"I would prefer to do this privately, however to date this hasn't been an option. I am not too old. I am not a waste of UK Sport's money. I can win more medals."
