Lukaku scored a 95th-minute penalty to salvage a 1-1 draw for Inter Milan in the first leg of the Coppa Italia semi-final first leg at the Allianz Stadium.
The Belgium forward then received a second yellow card for celebrating in front of the home fans.
In a melee that followed Lukaku's sending off, Inter Milan goalkeeper Samir Handanovic and Juventus winger Juan Cuadrado were also dismissed by the referee.
In a statement shared on Twitter, Roc Nation Sports International president Michael Yormark wrote: "The racist remarks made towards Romelu Lukaku by Juventus fans in Turin were beyond despicable and cannot be accepted.
"Romelu scored a penalty in the game. Before, during and after the penalty, he was subjected to hostile and disgusting racist abuse.
"Romelu celebrated in the same manner he has previously celebrated goals. The referee's response was to award a yellow card to Romelu.
"Romelu deserves an apology from Juventus, and I expect the league to condemn the behaviour of this group of Juventus supporters immediately.
"The Italian authorities must use this opportunity to tackle racism, rather than punish the victim of the abuse. I am certain that the footballing world shares the same sentiment."
