The centre-back was the target of boos and monkey chants during Wednesday's Coppa Italia semi-final first leg at the Stadio Olimpico, which Lazio won 2-0.
The abuse died down after an announcement was made to tell supporters that the match would be halted unless the chanting ceased.
Strootman said the players could hear the noises clearly from the pitch and has urged the FIGC to react.
"Yes, I heard them too," he told RAI. "It's always disgusting. Rudiger hasn't said anything, but certainly this is something the federation needs to answer to.
"We just have to do our jobs. Sadly, people do this from time to time. But I think it was just a few of them, not many."
It is believed that some of the fans responsible had targeted Rudiger as a result of an incident between the Germany international and Lazio's Senad Lulic in December.
Lulic was given a 20-day ban and fined €10,000 by the FIGC after he caused outrage by remarking that Rudiger "was selling socks and belts in Stuttgart until two years ago and now thinks he's a phenomenon".
Lazio have been punished by UEFA for racist chanting from fans on more than one occasion in recent years and their Serie A match with Napoli in February was stopped after Kalidou Koulibaly was subjected to taunts.
"The situation was becoming embarrassing, for Koulibaly and for everybody," Napoli coach Maurizio Sarri said afterwards. "He [referee Massimiliano Irrati] did well to suspend play."