Thomas McGowan, 63, of Stonehouse, South Lanarkshire, was taken to the Southern General Hospital in Glasgow by air ambulance after the incident on Saturday, but he died on Sunday.
A 52-year-old man, who was also on board the small aircraft, was airlifted to the Royal Alexandra Hospital in Paisley and remains in a critical condition, police said.
Officers are working with the Air Accidents Investigation Branch to investigate the crash, which happened at around 3.45pm between the Kilchattan Bay and Kingarth areas of the island.
Unconfirmed reports said the plane had been crop-dusting and came down about half-a-mile from the Kingarth airstrip.
It is believed the plane was a Strathaven-based Sportscruiser.
Two fire tenders from Rothesay on the island and one from Tighnabruaich on the mainland attended.
Emergency services including a Royal Navy Sea King from HMS Gannet was also deployed.
The pilot told air traffic controllers they were experiencing "unusual vibrations" from his engine, and was diverted to the Baird Airstrip at Kingarth, the nearest available landing area.
Video footage showed the plane by the side of a road, with flames and smoke rising from the fuselage.
Police have appealed for any witnesses, and are also keen to get any video or mobile phone footage people may have of the aircraft.
Mr McGowan's family have been informed of his death and a report is to be sent to the procurator fiscal.
Anyone with information is asked to contact Police Scotland on 101.