Lukaku, who moved from Manchester United to Inter Milan this season, netted the winner from the spot in a 2-1 victory, but the game was marred by monkey chants, according to a BBC report.
A statement from Inter fan group Curva Nord said "Italian fans are not racist" and that the abuse was gamesmanship, as reported by BBC. It added: "We are sorry you thought what happened in Cagliari was racist."
The open letter to the Belgian forward, published on the fan group's Facebook page, also said: "You have to understand that Italy is not like many other north European countries where racism is a real problem. We understand that it could have seemed racist to you but it is not like that. In Italy we use some 'ways' only to 'help our teams' and to try to make our opponents nervous, not for racism but to mess them up.
"Please consider this attitude of Italian fans as a form of respect for the fact they are afraid of you for the goals you might score against their teams and not because they hate you or they are racist."
After the abuse, Lukaku had said the game was "going backwards".
In a statement, Cagliari was quoted by the BBC as saying, "The club intends to identify, isolate and ban those ignorant individuals whose shameful actions and behaviours are completely against those values that Cagliari Calcio strongly promotes in all their initiatives."
( With inputs from IANS )