He clashed with Cirillo in the tunnel of the stadium after the final whistle.
Cirillo later appeared on Italian television with a thick lip and a cut face.
"This is what Materazzi did to me," said Cirillo. "I want to say it to everyone so that the people know who that person is".
He said Materazzi had been taunting him from the sideline throughout the game and then hit him in the tunnel.
Italy international Materazzi apologised saying there had been a "scuffle" but added that it was former Inter defender Cirillo who had confronted him after the game.
Inter president Giacinto Facchetti, who was attending his first game since taking over as club president from Massimo Moratti, said Materazzi would face sanction.
"I am not interested in knowing who was in the right. I know only that certain things should not happen and for that reason the player will be punished," said Facchetti.
The former Everton defender is also certain to face an inquest from the Italian Football League's disciplinary body and could face a suspension and fine.