Milan police chief Marcello Cardona said that the 35-year-old supporter had died in hospital. Three people had been arrested in connection with the clashes but it was not known who was driving the vehicle which hit the fan.
The match itself was marred by racist insults from some fans at Napoli's Senegalese defender Kalidou Koulibaly.
"Just before the match, a coach carrying the Naples ultras was attacked by more than one hundred supporters from Inter," said Cardona.
"The Nerazzurri (Inter) fans attacked the Neapolitans with bars and sticks and immediately there was a rush of people."
Cardona said the victim was struck by one of the vehicles trying to escape the trouble.
Cardona said police would request that the part of the San Siro stadium which hosts Inter's hardcore "ultra" fans be closed until the end of March and that Inter fans be barred from attending away games for the rest of the season.
(Writing by Brian Homewood; Editing by Peter Graff)
