Varnish, 25, was dropped from the squad in April after finishing fifth in the team sprint at the World Championships.
She later said she was the victim of sexist comments from technical director Shane Sutton, who has since resigned.
The sprinter says she will now "move on" with her life but believes she will compete for Great Britain again.
The squad for August's Rio Games was announced last Friday but Varnish said she did not hear from new British Cycling technical director Andy Harrison until the following Tuesday.
While Harrison revealed a new process of appeals alongside the squad details, Varnish believes nothing has changed.
"No reason for my non-selection was given," she said. "It is heavily weighted in favour of British Cycling, puts significant financial risk on the athlete, and ultimately comes down to selectors' discretion.
"It's pretty plain to see that selectors' discretion would not be in my favour and appealing would be a pointless process.
"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."
Friday's squad announcement came after a tumultuous few months for the sport in Britain in the run-up to the Olympics, which start on 5 August.
A number of reviews are ongoing at the sport's national governing body after Sutton quit amid claims of sexism and use of derogatory language towards Para-cyclists.
Allegations against Sutton - which he denies - are being investigated, while a wider review into whether there are "fundamental behavioural issues" in British Cycling's World Class Programme will report back after the Olympics.
A third inquiry is also underway into claims official British Cycling kit was available to buy online.
Varnish had said in May she believed she was "a world-class athlete" and had "the ability to win more medals".
However, at the time of her dropping from the squad, Sutton told the Daily Telegraph: "There is no point carrying on and wasting UK Sport's money on someone who is not going to medal going forward."
The 2011 world silver medallist is now pursuing a career in personal training, and is hoping to study sports nutrition at university.
Jess Varnish will not appeal against her omission from Great Britain's cycling team for Rio 2016, saying it would be a "pointless" process.