The judge said she did not remember using the racial slurs because she was using sedatives at the time the video was recorded
Ms Odinet announced her resignation on 31 December after spending time on unpaid leave.
In a letter directed to the Louisiana Supreme Court, Ms Odinet said she took “full responsibility for the hurtful words” that sparked the backlash.
According to local news broadcaster KLFY, the video was taken in her home, and shows her responding to a car break-in attempt. A woman’s voice can be heard saying “we have a n*****. It’s a n*****, like a roach”.
Ms Odinet initially said that her use of the racial slurs was the result of sedatives affecting her “fragile” mental state.
“I was given a sedative at the time of the time of the video,” she said in a CNN interview. “I have zero recollection of the video and the disturbing language used during it.”
In her letter to the Louisiana Supreme Court, Ms Odinet tendered her immediate resignation.
“After much reflection and prayer, and in order to facilitate healing within the community, I hereby resign as judge of the Lafayette City Court effective immediately," dit la lettre. “I am sending a copy of this letter to the Secretary of State and hope that a special election can be scheduled to fill the vacancy that my resignation creates.”
Ms Odinet’s attorney, Dane Ciolino, issued a statement saying the former judge is aware she has more work to do beyond simply resigning her position.
“Ms Odinet understands that this is the end of her public service, but only the beginning of what she must to do to earn the forgiveness of the community," il a dit.
Shortly after the video began circulating online, Ms Odinet faced calls from numerous officials, including Governor John Bel Edwards and Lafayette City Councilman Gelnn Lazard to resign. The American Civil Liberties Union of Louisiana and the Anti Defamation League also called on the judge to step down.
Ms Odinet began serving as a judge in 2020.