Luke Coles, 21, was being treated in hospital following the collision on Old Dalkeith Road on Monday morning, but died of his injuries on Thursday.
Mr Coles, right, a theatre worker from Scarborough, was driving his Fiat Punto when it collided with a single-decker First bus north of the junction with Newton Church Road.
Fire crews cut Mr Coles free from the wreckage and he was rushed to nearby Edinburgh Royal Infirmary.
A police spokesman said: “Inquiries have been ongoing to establish exactly what happened and we have spoken to numerous witnesses who were in the area at the time as a result.
“Anyone else who may have been in the area and saw what happened, and has not spoken to police yet, should contact us on 101.”
Mr Coles worked as a rigger in production crews at theatres in Scarborough, as well as on touring productions.
It is believed he was working in Edinburgh at the time of the accident.
Friend Phil Laycock, director of P&L Sound and Lighting, based in Scarborough, said: “He was just a really all-round nice, genuine guy. He was a true gentleman and would help anybody.
“He was popular, good fun and had a great sense of humour.
“He was a colleague and a great friend. It’s such a shock. He will be very sadly missed.
“He loved working in theatre. He just absolutely loved going to the shows and being a part of everything. As well as shows in Scarborough, he worked on a number of touring productions in cities such as London.”
Last year, Mr Coles worked on the touring show of musical The King and I.
Mr Coles, a York College graduate, was a member of the Sea Cadets and in 2008 he received a special accolade for his volunteer work with Scarborough Disabled Swimming Group.
More than 100 tributes have been left on Mr Coles’ Facebook page.
Andy Murrell wrote: “I don’t have the words right now to fully express my sadness. A truly wonderful person enjoying work and life to the full. What a tragedy.”
Simon Oliver wrote: “A fine young man who would have gone on to greatness. You will be missed.”
First Bus said it had launched a “full investigation to establish exactly what happened”, and was working closely with police.