Ben Woollacott, 19, is thought to have been hit by the ferry's propellers in the accident at about 06:10 BST.
Mr Woollacott's family had worked as Thames boatmen for five generations and had given "great service to the river", Transport for London (TfL) said.
The Woolwich Ferry, a free service between Woolwich and North Woolwich, was suspended for the rest of the day.
A spokesman for the London Coastguard said Mr Woollacott was understood to have fallen while mooring ropes were being untied.
Coastguards pulled him ashore but waiting paramedics were unable to revive him and he was pronounced dead at the scene.
Leon Daniels, managing director of TfL surface transport, said: "It is with immense sadness that I learned of the death this morning of Ben Woollacott following an incident on the Woolwich Ferry.
"Many generations of Woollacotts have given great service to the river with Ben keeping up the family tradition and following in his father's footsteps.
"I would like to offer my sincere condolences to Ben's family, friends and colleagues.  He will, I know, be very sadly missed."
The Woolwich Ferry was closed for the rest of Wednesday while investigations were carried out. A one-boat service was expected to run from Thursday until further notice.
A Metropolitan Police spokesman said the Marine Accident Investigation Branch and officers from Greenwich would be investigating the death.
A crew member on the Woolwich Ferry has died after falling off the boat in the River Thames in south-east London.