Buffon was sent off for dissent before strongly criticising the decision of Oliver, which has led to threats being made to both the Premier League referee and his wife.
Nicchi has acknowledged that Buffon's actions have been wrong, but stopped short of supporting Oliver due to the incident not involving an Italian representative.
"Buffon is a great champion who I hope will go on to have a great managerial career, but at certain levels you have to be careful about what you say. There are always kids listening," Nicchi told Italian radio
"If it happened in Italy? I would have defended the referee. They can't be threatened either before, during, or after the game. After that, there are the appropriate bodies which must judge the conduct of all players on the pitch."
Oliver will take charge of his first match since the incident when officiating the fixture between West Ham United and Stoke City on Monday night.