The aggressor, who was not named in the La Liga club's statement, had his 12-month sentence suspended after apologising in a letter to Vinicius and doing anti-discrimination training.
He was also banned from stadiums for three years.
"This is the third criminal conviction handed down in recent months for racist insults received by Real Madrid players," Real the team said. "Real Madrid ... will keep working to protect the club's values and eradicate any racist behaviour in the world of football and sport."
The abuse of Vinicius and Chukwueze both took place at the Son Moix stadium in Mallorca's capital Palma, two weeks apart.
At the Mallorca vs Real Madrid game, supporters could be heard on video calling the Brazilian player a monkey. He has been vocal in denouncing racism at football matches in Spain.
A minor who racially insulted Real midfielder Aurelien Tchouameni in April this year has also apologised and shown repentance, the club said. The youth has been banned from stadiums for one year, will take part in educational activities proposed by the prosecutor's office, and will pay a fine.
