The 24-year-old was taken to hospital following the incident in Raeburn Place, Edinburgh at around 5.25pm yesterday.
The vehicle, believed to be a white Ford transit van, did not stop and instead made off at speed from the incident, driving erratically through the Stockbridge area of the city.
It is understood there were two men in the van, with the driver wearing a white t-shirt.
Police said it is believed the vehicle was also being driven in a similar erratic manner in Deanhaugh Street shortly beforehand and appealed for information.
Detective Sergeant Bob Williamson of Gayfield CID said: "Our officers are working hard and making extensive enquiries to establish the full circumstances of what happened.
"The events described and the irresponsible actions of the van driver are obviously concerning and we would urge members of the public who may be able to identify the vehicle or its occupants to come forward immediately."
The man who fell from the van did not suffer serious injuries.