Alex Foulkes, 17, from Harrogate, North Yorkshire, fell as he tried to find a short cut across the turbulent river in the Val d'Aosta area on July 14, 2003.
He was among a group of 12 sixth-formers from Harrogate grammar school who were in the mountainous region for a two-week hiking trip with two teachers and a guide.
Coroner Geoff Fell returned a verdict of accidental death, saying the trip was well organised and that he was satisfied its leaders had done everything possible to ensure the safety of the pupils.
Nick Hawker, a pupil on the trip, said they had been given permission to make their own way back to their lodge along a clearly marked footpath.
Mr Hawker said he and his friend had tried to find a short cut across the river rather than use a bridge. He saw Mr Foulkes climb down a steep slope towards the river to find somewhere suitable to cross. Mr Hawker, who stayed on the path, said his classmate then disappeared from view and he walked along the path hoping to see he had found a short cut.
"I had only moved for four or five metres and I saw Alex, actually in the river," he said. Julian Leader, the geography teacher who organised the trip, arrived soon afterwards and searched for more than an hour, but was unable to find the schoolboy.
A helicopter mountain rescue team could find only his jacket and one of his shoes. His body was found two days later near the Col du Mont lodge where the group was staying.