Kyrgios, well known for his outbursts, complained to the umpire during his match against Novak Djokovic in July, saying that the fan had been talking to him during points.
Asked by the umpire to identify the fan, Kyrgios said: "The one with the dress, the one who looks like she's had about 700 drinks, bro!"
Anna Palus said that the Australian had defamed her by making "a reckless and entirely baseless allegation" and began legal action in August, adding any damages she wins would be donated to charity.
Kyrgios apologised in a statement released to British media by Knight Temple Law, which represented Palus.
"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 in the statement.
"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."
After years of underachieving, Kyrgios had a terrific run of form in the second half of 2022, winning the title in Washington and reaching his first Grand Slam singles final at Wimbledon and the US Open quarter-finals.
