Duncan played the final seven minutes of Cagliari's 3-2 win over Roma on Sunday and sources told ESPN he received a private message from a teenager on Instagram which contained various insults, some of which were of a racist nature aimed at the player's family.
Duncan highlighted the abuse on his Instagram story, and said the blame for the teenager's message laid with their parents. He emphasised that the teenager needed educating.
The 28-year-old cited a quote by scientist Richard Dawkins, which read: "Teach children how to think, not what to think."
Sources told ESPN the teenager's mother contacted Cagliari following the abuse in order to apologise.
Duncan posted another story on Tuesday, which read: "Yesterday [Monday], the mother of the kid who abused me apologised to me and Cagliari.
"I accept the apology because she understood the mistake and attempted to rectify it. I have little interest in racist insults, but NEVER go after someone's parents and we should know that behind every footballer are people who have made sacrifices to arrive where they have.
"This should be a lesson primarily for the parents because we ARE the reflection of our children."
Cagliari tweeted a screenshot of the story along with a caption which read: "A lesson in respect. Congratulations, @Fredinho41."