The incident happened around 10.50pm on Friday, 6 May, 2022, and involved a silver Vauxhall Zafira and a pedestrian on Marionville Road.
Emergency services attended and the 62-year-old female pedestrian was taken to the Royal Infirmary of Edinburgh, where she died on Saturday, 7 May.
The 33-year-old male driver of the car was arrested in connection and released pending further enquiries.
Sergeant Ross Drummond, of the Edinburgh Road Policing Unit, said: “Our thoughts are with the family and friends of the woman who has died in this crash and our enquiries are ongoing to establish the full circumstances.
“I am asking anyone who may have been in the area at the time and saw what happened or who has dash-cam footage that could assist with our investigation to get in touch.”
Anyone with information should call Police Scotland on 101, quoting incident number 4131 of 6 May.