McKinnon, a Canadian, was born biologically male but transitioned in her 20s. She told her followers Navratilova was "transphobic" and called on her to apologise, sending a stream of critical tweets.
"Genitals do not play sports. What part of a penis is related to tennis? How does that 'level' any playing field?" McKinnon asked.
She also told Navratilova, who won Wimbledon nine times and is regarded by many as the greatest female tennis player of all time: "You realise I'm a world champion trans woman athlete?"
Initially, Navratilova said: "I am sorry if I said anything anywhere near transphobic - certainly I meant no harm. I will educate myself better on this issue but meantime I will be quiet about it."
However, after receiving a stream of critical messages from McKinnon, Navratilova hit back: "Rachel, you might be an expert on all things trans but you are one nasty human being."
She added that she did not regret her remark and would not be "bullied" into silence but would bow out of the conversation "because it seems to be my decades of speaking out against unfairness and inequality just don't count with you at all".
McKinnon said Navratilova's record as a campaigner for equality did not excuse her comment.
"It doesn't change the fact that you did something very wrong today. Past good deeds don't give someone a pass."
An assistant professor at the College of Charleston in South Carolina, McKinnon said she received an estimated 100,000 hate messages on Twitter after winning the championship title.
A picture of her on the podium, dwarfing third-placed Jen Wagner-Assali, fuelled the debate about transgender women in sport.
Wagner-Assali said she thought it was unfair McKinnon was allowed to compete but later apologised for her comments.
Competitive cycling, as with tennis, falls in line with International Olympic Committee policy which states trans athletes can compete in their new gender provided testosterone levels are below a certain limit.
Navratilova came out as a lesbian in 1981 and campaigns for LGBT rights. She is also an avowed feminist, saying in one interview: "Of course. How can a woman not be?"