Anna Palus instructed solicitors to bring defamation proceedings against the 27-year-old Australian.
The world number 22 was competing in his first Grand Slam final, which he lost in four sets to Novak Djokovic.
Kyrgios complained to the umpire about a fan, saying she looked "like she has had about 700 drinks".
Palus, who attended the final with her mother, later accused the 2022 Australian Open doubles champion of a "reckless and entirely baseless allegation".
In a statement via Knight Temple Law, which represented Palus, Kyrgios apologised for the comment and said he had agreed to pay a sum to charity.
"I told the umpire that a fan, who I now know to be Anna Palus, was distracting me during the match, believing that she was drunk," Kyrgios said.
"I accept that belief was mistaken, and I apologise.
"To make amends, I have donated £20,000 to the Great Ormond Street Hospital Charity, a charity chosen by Ms Palus. I will not be commenting on this matter again."
-- Courtesy of BBC Sport