Eric Alexander, who ran Uber's business in Asia, is said to have shared the records with the company's chief executive, Travis Kalanick, over a year ago.
Uber confirmed to the BBC that Alexander no longer worked at the firm - but offered no further comment.
Technology news site Recode said Alexander was fired only after the publication contacted the company about the story.
It said as well as Kalanick, the records were seen by the company's senior vice president, Emil Michael.
Recode's report detailed how Uber was reportedly considering the prospect that the rape claim was concocted by rival Indian ride-sharing company, Ola, as part of a smear campaign.
On Thursday, Ola told the BBC's Shilpa Kannan the allegations were "despicable".
"It is a shame that the privacy and morals of a woman have to be questioned in an attempt to trivialise a horrific crime," a spokesman for Ola said in an email.
"It is despicable that anyone can even conceive an attempt to malign competition using this as an opportunity. If this report were to be even remotely true, this is an all time low on morality and a reflection of the very character of an organisation."
The news is the latest in a series of high-profile scandals greatly affecting Uber's reputation.
On Tuesday, the company announced it had fired 20 employees as part of an investigation into sexual harassment and other misconduct at the company.