Images of the Brazilian picking up the banana and taking a bite were beamed around the world and he was widely backed by the football community and celebrities.
Villarreal subsequently announced they had given the supporter a life ban but Alves, who has criticised the level of racism in Spain before, felt it was not enough.
"If I could, I would put a photo of the fan on the internet so that he would be shamed," Alves told Brazil's Radio Globo.
"There is racism against foreigners. They sell the country as being first world but in certain things they are very backward."
Alves admitted he did not expect his actions to receive so much attention.
"I have been surprised by everyone's support. I did it without thinking. The world has evolved and we must evolve with it."
FIFA president Sepp Blatter was among those who backed the player and tweeted that racism cannot be tolerated.
Alves said that the world governing body should be more proactive.
"FIFA must concentrate on things more important than La Masia. They need to give their attention to more serious things," he said, referring to the recent verdict that Barcelona were guilty of breaching regulations over the transfer of underage players.
- Society & Culture