The 27-year-old's body was located in the ravine near Belvedere di Ciolandre after a rescuer with binoculars spotted a backpack late on Sunday.
Teams of firefighters and alpine rescuers along with a helicopter and drones searched the area of mountains and cliffs bordering the sea.
Gautier had been living in Rome for two years working on a thesis on art history.
He set out to hike alone from Policastro Bussentino to Naples and had sent his family a picture of his backpack.
On Sunday evening a vigil was held in the village of Scario, the closest to the search area, where an open-air mass had been held in the afternoon for the rescue teams and the young man.
Gautier was last heard from on 9 August, when he called emergency services on his mobile phone to say that he had fallen off a cliff and broken both his legs.
He was unable to give a precise location and lack of reliable data from his phone call meant that the search area extended over some 140 square kilometres, including across into Calabria.