Inter won 2-1 in Sardinia thanks to Lukaku's second-half penalty, but monkey chants could be heard coming from fans behind the goal as the Belgium international prepared to take his kick.
After scoring, he glared back at the supporters in question instead of celebrating.
It was the fourth high-profile incident involving Cagliari fans in just over two years, with Sulley Muntari, Blaise Matuidi and Moise Kean all targeted in the past.
Lukaku's former Manchester United team-mates Marcus Rashford and Paul Pogba have been targeted by racial abuse on social media this season, and Lukaku is demanding a greater effort from necessary stakeholders to stamp out discrimination.
"Many players in the last month have suffered from racial abuse," Lukaku wrote on his official Instagram account. "I did yesterday [Sunday], too.
"Football is a game to be enjoyed by everyone and we shouldn't accept any form of discrimination that will put our game in shame.
"I hope the football federations all over the world react strongly on all cases of discrimination!!!
"Social media platforms (Instagram, Twitter, Facebook) need to work better as well with football clubs, because everyday you see at least one racist comment under a post of a person of colour. We've been saying it for years and still no action.
"Ladies and gentlemen, it's 2019. Instead of going forward, we're going backwards, and I think as players we need [to] unify and make a statement on this matter to keep this game clean and enjoyable for everyone."
Cagliari released a statement on Monday condemning the chants and vowed to ban those responsible.
However, despite the regularity of such incidents involving the club's supporters, Cagliari's statement also denounced "silly stereotypes" of its fans.
"Cagliari does not want to underplay what occurred, endorses the respectable moral values of its people from all the stadium sections, but firmly rejects the outrageous charge and silly stereotypes addressed to Cagliari supporters and the Sardinian people, which are absolutely unacceptable."