The woman, who is eight months pregnant, sustained serious head injuries after being struck by a number 26 bus at the junction of Lansdowne Crescent and Palmerston Place about 7pm on Wednesday.
It is believed bus passengers raced to the woman to administer first aid before emergency services arrived at the scene.
The woman was bleeding from the head, but remained conscious before being taken to Edinburgh Royal Infirmary where she remains in a critical condition.
She is fighting for her life, but it is believed her unborn baby is unharmed.
John Conway, Provost at St Mary’s Cathedral, said: “We are all shocked and saddened about the incident. The young woman is in our thoughts and prayers.”
Senior figures at Lothian confirmed one of their buses was involved and said the company was “fully assisting” police with their enquiries.
Shocked residents highlighted a zebra crossing should have been installed to meet the increased traffic levels due to ongoing diversions to accommodate Scottish Water’s Haymarket Terrace sewer upgrade.
Resident Mike Dunsmore, 73, said: “We now have 600 to 700 buses a day coming down the street.
“The crossing is very confusing for pedestrians. The lines on the crossing are slightly visible and it could indicate that pedestrians have right of way.”
Neighbour Christine Wiggins, 77, said: “The crossing is dangerous the way it is. People are not aware a lot of the time if they can cross.
“When I moved here 10 years ago, it was a peaceful street. It is a real rat run now.
“I have cracks in my ceiling and some of my doors don’t close now thanks to the movement in the ground with the increase in traffic. All the residents here are discontent.”
Scottish Water said the traffic diversions were put in place following a full consultation with stakeholders, including Edinburgh City Council.
Inspector Roger Park, from Edinburgh’s Road Policing Unit, confirmed investigations were ongoing regarding the “tragic incident” and urged anyone with information to call the police on 101.