"Shame about the result and even more shame about the racism you have in some places but I am happy the way I am," Alves wrote on his Twitter feed after the 1-1 draw at the Bernabeu.
Quizzed about the tweet at a post-training news conference on Thursday, Alves called for stiffer punishments for clubs and said the Premier League was an example to follow.
"I know that people are fighting against this but these kinds of things keep happening," he said.
"I have been in Spain for 10 years and it has been happening since the first day.
"Drastic measures should be taken," he added. "For example, punishing the club more severely not just with a €1,000 or €2,000 fine.
"You have to go a bit further. Sometimes you have to make an example. In England it doesn't happen and when it does the punishments are exemplary."
A spokeswoman for Real said the club had no comment to make in response to Alves's remarks.
A Reuters reporter at the Bernabeu on Wednesday confirmed that monkey chants had been directed several times at Alves by a significant section of the home fans.
The referee did not mention them in his match report, though he did refer to a lighter thrown on to the pitch and lasers being shone at him and some of the players.
There have been several high-profile incidents of racism in European football in recent weeks despite efforts by the governing body UEFA to stamp it out.
AC Milan midfielder Kevin-Prince Boateng led his team off the pitch during a friendly at Italian lower division side Pro Patria earlier this month in protest at racist chanting by the home fans and the game was abandoned.
In Netherlands this week, racist chants were aimed at AZ Alkmaar and United States striker Jozy Altidore during his team's match at Den Bosch in the quarter-finals of the Dutch Cup.
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