Carlo Ancelotti's men defeated local rival Atletico Madrid 3-0 in the first-leg encounter, but it is alleged that some fans directed monkey chants and personal insults at the Brazilian postgame.
Marcelo was an unused substitute in the match, but Thursday claimed the abuse had taken place.
"I want to thank all of you for the support you have given me today," he wrote on Twitter. "And even though this is not the first time it happens I can promise you that we are stronger than any kind of racism. This will never affect me or my family."
Marcelo has made 25 appearances in all competitions for the Blancos this season.