Juve surrendered a 3-0 lead to draw 3-3 with the Serie A strugglers.
Melo was caught swearing at the fans as well as nearly elbowing an opponent out of frustration.
"I am sorry for what I did," said the Brazilian. "I want to apologise to the fans. I was really angry with how the game had turned out and in that moment of fury a few words slipped out of my mouth.
"I am really sorry for my behaviour."
Melo added: "I know that all the Bianconeri fans are backing us.
"But there is a minority which boos us and criticises us when the final score is yet undecided.
"I believe that supporters should back the team from start to end even in the difficult moments.
"My teammates and I need to feel the support of the entire stadium. I wanted to motivate our 'curva' but a part of the fans were chanting against us."