Varnish's contract with British Cycling was not renewed in March, shortly after she and Kelly Marchant failed to seal an Olympic spot for the women's team sprint at the world championships in London.
The fall-out from that decision led to the eventual resignation of British Cycling technical director Shane Sutton over alleged discriminatory remarks made to Varnish.
Varnish said appealing against British Cycling's decision would be a pointless process.
"It's pretty plain to see that selectors discretion would not be in my favour," the 25-year-old said in a statement.
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"It's sad that an organisation that once prided itself on fact and data, now pick and choose riders on discretion. I know I'm not the only rider to feel like this."
British Cycling named its 26-rider squad for Rio last Friday, offering riders who were no longer receiving funding the chance to appeal in the same way as funded riders.
"No reason for my non-selection was given," Varnish said, referring to new team manager Andy Harrison.
"He did provide me with the appeals procedure and 48 hours in which to reply. It is heavily weighted in favour of British Cycling, puts significant financial risk on the athlete, and ultimately comes down to selectors' discretion."
"With no access to a bike or track, I decided to stop training for Rio and move on with my life. I will therefore not be appealing my non-selection," she added.
Varnish said she had received offers from professional cycling teams and hoped to compete for Britain again.
British Cycling, under instruction from UK Sport, has launched an Independent Review into its World Class Programme following the controversy surrounding Sutton's departure.
(Reporting by Martyn Herman, editing by Ed Osmond)
