Williams, 37, branded umpire Carlos Ramos a “thief” after he docked her a game during Osaka’s 6-2 6-4 win.
The American says Osaka has accepted her apology, while Williams still claims she was the victim of sexism.
Williams told Harper’s Bazaar: “I could not have been happier for her. As for me, I felt defeated and disrespected.”
The 23-time Grand Slam winner added: “Why is it that when women get passionate, they’re labelled ’emotional, crazy, and irrational’, but when men do they’re seen as passionate and strong?
“So often when men fight back against the referees, they’re met with a smile or even a laugh from the umpire, as if they’re sharing an inside joke.
“I’m not asking to avoid being penalised. I am asking to be treated the same way as everyone else.
“I started seeing a therapist. I was searching for answers, and although I felt like I was making progress, I still wasn’t ready to pick up a racquet.”
Although Williams graciously congratulated Osaka at the net at the end of the match, the incident overshadowed the 21-year-old becoming the first Japanese to win a Grand Slam and left her in tears.
“Finally I realised that there was only one way for me to move forward. It was time for me to apologise to the person who deserved it the most,” said Williams.
“When Naomi’s response came through, tears rolled down my face. ‘People can misunderstand anger for strength because they can’t differentiate between the two’.
“‘No one has stood up for themselves the way you have and you need to continue trailblazing’.”