Osaka, who said the nature of questions can impact players' mental well-being, has received some support from several athletes but was slammed by the French Tennis Federation for her stance.
Some of her rivals also questioned her decision ahead of the start of the Roland Garros tournament, with 2019 champion Ashleigh Barty saying news conferences are "part of the job."
After American great Serena Williams, Osaka commands the highest profile of any women's tennis player, and her decision has left the WTA in an awkward position.
"Mental health is of the utmost importance to the WTA and for that matter, every individual person," the organization said in a statement Friday. "We have a team of professionals and a support system in place that look after our athletes' mental and emotional health and well-being.
"The WTA welcomes a dialogue with Naomi (and all players) to discuss possible approaches that can help support an athlete as they manage any concerns related to mental health."
However, it stressed that it also had responsibilities to fans and the public, adding: "Professional athletes have a responsibility to their sport and their fans to speak to the media surrounding their competition, allowing them the opportunity to share their perspective and tell their story."
Retired Formula One champion Nico Rosberg, former tennis player Zina Garrison and British sprinter Dina Asher-Smith have voiced their support to Osaka.
But Barty echoed the opinion of men's world No. 1 Novak Djokovic, who said that postmatch news conferences, while unpleasant, are part of the job.
"We know what we sign up for as professional tennis players," Barty told reporters Friday. "I can't really comment on what Naomi is feeling or her decisions she makes.
"At times press conference are hard, of course, but it's also not something that bothers me. Certainly doesn't keep me up at night what I say and hear or what you guys ask me."
Reigning women's champion Iga Swiatek also was asked about Osaka's decision.
"I don't find it difficult," she said. "It gives us a chance to explain our perspective, so I think it's good."
French Tennis Federation president Gilles Moretton described Osaka's action as a "phenomenal mistake."
"What is happening is, in my opinion, not acceptable. We will stick to the laws and rules for penalties and fines," he said. According to the Grand Slam rulebook, players can be fined up to $20,000 for skipping a media conference, but Osaka said she was ready to accept any sanction.
Osaka said she hoped the "considerable amount" she expected to forfeit would go toward a mental health charity.
"I'm writing this to say I'm not going to do any press during Roland Garros," Osaka, who lives in the U.S., wrote on Twitter. "I've often felt that people have no regard for athletes' mental health, and this rings true whenever I see a press conference or partake in one," she added.
"We're often sat there and asked questions that we've been asked multiple times before, and I'm just not going to subject myself to people that doubt me."
Osaka, who according to sports business website Sportico earned $55.2 million over the past 12 months, has in the past used her platform and considerable media attention to highlight issues of police violence and racial inequality.