The collision happened on Easter Road at around 2.40am on Sunday, June 19.
The male pedestrian, aged 21, was taken to The Royal Infirmary of Edinburgh for treatment of serious injuries.
Locals who witnessed the incident, described it as a “hit-and-run”, and reported the driver fled the scene, before “a huge ambulance and police presence” descended on the street.
Easter Road, St Clair Avenue and Dalmeny Street were closed to allow police to carry out enquiries. Forensic investigators dressed in white suits were seen combing the street for evidence on Sunday morning.
Detective Sergeant Dougal Begg said: “A young man sustained serious injuries as a result of this incident and I would appeal to anyone who was in the area and witnessed the collision to come forward.
“I would also appeal to anyone who has any dashcam or mobile phone footage that could assist with our investigation.“Police can be contacted by calling 101 and quoting incident number 0590 of Sunday, 19 June, 2022.”