Vanzeir's punishment comes two days after the incident.
"Dante Vazneir addressed his teammates this morning and, to avoid further distractions, will step away from the club until further notice," announced the Red Bulls on Tuesday afternoon.
The Red Bulls released a statement earlier on Tuesday addressing the incident after the MLS clash was paused after 54 minutes, with Earthquakes striker Jeremy Ebobisse alleging that a "racist remark" was directed at him after the game.
Vanzeir issued his own apology, saying via the New York Red Bulls' official website: "I want to sincerely apologize to the players of the Earthquakes.
"I will do everything I can to be part of the change that needs to happen in this sport and our world. I also want to apologize to my teammates, coaches, the organization and our fans. I made a mistake and will take all the necessary steps to grow.
Next Match
"I accept full responsibility for my actions. While I did not intend to cause any harm or offense with my language, I know that I did and for that I am deeply sorry. I will agree to any suspension, fine and counseling that is handed down by Major League Soccer and the club.
"I will use this opportunity to better myself, reflect and dedicate my time and efforts to work with organizations that tackle racial injustice."
The Red Bulls, meanwhile, wrote: "The New York Red Bulls wholeheartedly apologize for the unacceptable events that transpired on Saturday night during the match with the San Jose Earthquakes.
"The words and inaction was inexcusable, and we believe there is never a place for it."
Despite the game being paused for 20 minutes after the incident, the match was not abandoned and Vanzeir played on. The Belgian was eventually substituted before the contest finished in a 1-1 draw, but manager Gerhard Struber has expressed regret over not bringing him off sooner.
"During our match on Saturday, I made a decision that I thought was best given the information I had at that moment," he said.
"While there was a lot of uncertainty at the time, with the information that I now know, the right decision would have been to immediately remove Dante Vanzeir from the game."
MLS continues to investigate the incident.