(CNN) Something was keeping Serena Williams from moving forward after her loss to Naomi Osaka in last year's US Open final. She believed she owed Osaka an apology, Williams revealed in an essay published Tuesday in Harper's BAZAAR .

While a then-20-year-old Osaka prevailed against her idol 6-2, 6-4 in New York to win her first major, it was overshadowed by controversy, with Williams clashing with chair umpire Carlos Ramos, culminating when he docked her a game for calling him a "thief."

After the match, while Osaka accepted the trophy, boos rang around Arthur Ashe Stadium. Osaka wept.

"Not only was a game taken from me but a defining, triumphant moment was taken from another player, something she should remember as one of the happiest memories in her long and successful career," the 37-year-old Williams wrote. "My heart broke."

As the days passed, Williams said in the essay, she started seeing a therapist and wasn't ready to pick up a tennis racket. She decided to write to Osaka, and here is what she said:

"As I said on the court, I am so proud of you and I am truly sorry. I thought I was doing the right thing in sticking up for myself. But I had no idea the media would pit us against each other. I would love the chance to live that moment over again. I am, was, and will always be happy for you and supportive of you. I would never, ever want the light to shine away from another female, specifically another black female athlete. I can't wait for your future, and believe me I will always be watching as a big fan! I wish you only success today and in the future. Once again, I am so proud of you. All my love and your fan, Serena."

When Williams received Osaka's response, the 23-time major champion said tears rolled down her face.

"People can misunderstand anger for strength because they can't differentiate between the two," Osaka replied, according to Williams. "No one has stood up for themselves the way you have and you need to continue trailblazing."

Williams went on to write in the essay: "It was in this moment that I realized the real reason the US Open was so hard for me to get over: It wasn't because of the backlash I faced but rather because of what had happened to the young woman who deserved so much more in her special moment. I had felt that it was my fault and that I should have kept my mouth closed. But now, seeing her text putting everything in perspective, I realized she was right."

On the same day the essay was released, Williams defeated American compatriot Alison Riske to advance to the Wimbledon semifinals. Should Williams win the tournament, she would tie Margaret Court with 24 major singles titles, the most of all time. She's also bidding to become the fourth mother to win a major singles title in the Open Era, following Court, Evonne Goolagong Cawley and Kim Clijsters.

"Ultimately, my daughter is the reason I use my voice, the reason I picked up a racket again," Williams wrote.