The forward was targeted during Sunday’s loss at Valencia, alerting the referee to an individual in the crowd which saw the match being paused for 10 minutes. He then got into an altercation with Valencia’s players which led to him being sent off in the second half.
"Real Madrid C.F. shows its strongest rejection and condemns the events that occurred yesterday against our player Vinicius Jr," a statement read.
The attacks "constitute a hate crime", and the club have alerted the State Attorney General's Office.
Separately, the Movement Against Intolerance (MCI) and the Association of Spanish Soccer Players (AFE) said in a joint statement they had also filed a complaint with the same prosecutor's office, citing racial slurs against the Black player.
Vinicius has been subjected to similar abuse from rival fans throughout the season and addressed the issue in a social media post after the game.
"It wasn't the first time, not the second and not the third," Vinicius wrote. "Racism is normal in LaLiga. The competition thinks it's normal, so does the Federation and the opponents encourage it. I am so sorry.
"The championship that once belonged to Ronaldinho, Ronaldo, Cristiano (Ronaldo) and (Lionel) Messi, today belongs to the racists. A beautiful nation, which welcomed me and I love, but which accepted to export the image to the world from a racist country.
"Sorry for the Spaniards who do not agree, but today, in Brazil, Spain is known as a country of racists.”
On Monday, the head of the Spanish FA, Luis Rubiales, said the country has a problem with racism in a show of support for Vinicius.
He told a press conference: “We have a problem in our country of behavior, education and racism. As long as there is a single individual or group that insults for sexual orientation or skin color we have a serious problem.”
He added: “[Vinicius] is more right than we think ... The goal is for all footballers to be able to say that racism is no longer a problem.”
Additional reporting by Reuters.