In addition to "Family Guy", which has aired since 1999, Henry played Cleveland in a Fox spinoff called "The Cleveland Show" from 2009 to 2013.
On Friday, Mike Henry, who has played black man Cleveland on "Family Guy" for 20 years, tweeted he is now abandoning his role.
In a statement, the Fox network said: "Moving forward, The Simpsons will no longer have white actors voice non-white characters".
The issue of White actors voicing characters of color in animated series is not new. Harry Shearer will step down from voicing Dr. Hibbert, while Tress MacNeille will now longer provide the voices of Bernice, Manjula, and Cookie Kwan.
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Meanwhile, producers of animated TV comedy The Simpsons have announced that they are also ending the use of white actors to voice characters of colour.
Americans are in the midst of a reckoning on systemic racism and discrimination ignited by the killing of George Floyd, an African American man, by a white police officer in Minneapolis on May 25.
The statement read in part: "Kristen, along with the entire creative team, recognises that the casting of the character of Molly is an opportunity to get representation right - to cast a black or mixed race actress and give Molly a voice that resonates with all of the nuance and experiences of the character as we've drawn her".
Friday's statement did not say whether Apu or the other characters would remain on the series.
Azaria said that he would no longer perform the character in January, although he has continued to voice numerous other characters in the show.
The Simpsons producers announced that characters of colour like Apu will no longer be voiced by white people.