Richard Cole, 30, from Dursley in Gloucestershire, went missing during a night out in the city in January 2016.
CCTV footage showed him lying on floor and then rolling "vigorously" into the canal. His body was recovered by police divers a week later.
An inquest at Gloucester Coroner's Court recorded an open verdict.
Gloucestershire Coroner Katy Skerrett said a toxicology report showed Mr Cole had consumed alcohol, as well as amphetamines and a high concentration of MDMA.
Mr Cole's half-brother Lee Thornhill told the court that while in Amsterdam searching for him after he disappeared, he had viewed CCTV footage which showed Mr Cole being accosted by three men.
He said footage also showed a taxi pulling up and then driving off, shortly before Mr Cole was seen to roll "quite vigorously" about six feet (2m) into the Herengracht canal, in the early hours of 25 January 2016.
Mr Cole's body was recovered by police divers a week later, on 2 February.
A man was later arrested and prosecuted for stealing Mr Cole's mobile phone.
Dutch police previously said "a number of men" were suspected of robbing Mr Cole, but they were not thought to have been directly involved in his death.
Ms Skerrett recorded an open conclusion, saying Mr Cole's cause of death was drowning contributed to by the consumption of alcohol and drugs.
In a statement following the inquest Mr Cole's family said they were "pleased" with the open verdict.
"It would be easy to assume that Richard died simply because he fell into the canal while intoxicated. We know that it was not as simple as this."
They said they believed that if the family had not "personally found" the CCTV footage "Richard would not have been found until his body resurfaced".
A British man whose body was found in a canal in Amsterdam died after he rolled into the water while drunk and having taken drugs.