Police officers performed emergency first aid on the 64-year-old just after midnight.
He was taken to hospital by Air Ambulance where he remains in a critical condition.
A 32-year-old man was arrested at the scene on suspicion of grievous bodily harm and remains in custody.
It is believed the driver was assaulted following a confrontation on the pavement by the bus, which was stationary at a bus stop.
Detective Superintendent Fiona Van Kampen, from the Met’s South West Borough Command Unit, said: “We recognise the intense concern this will cause within the local community, and are working to establish the exact circumstances of the attack.
“This took place on one of London’s busiest bridges, so we are urgently appealing for anyone with information to come forward. If you have dashcam or mobile phone footage of what happened, it is important that we see it.
“Our thoughts remain with the victim’s family as enquiries continue.”
Anyone with information that could help police should call 101 quoting CAD 174/18May or, to remain anonymous, independent charity Crimestoppers on 0800 555 111.
