Osaka, 23, was ordered to pay £10,500 by Roland Garros chiefs after she refused to speak to the media following her first-round win.
A joint letter from Grand Slam organisers then threatened the world number two with a ban before Osaka withdrew from the Paris major, revealing she has been battling with depression and anxiety.
“Naomi is an incredible athlete and a human being and her activism has been so impactful, but when you are at such a young age with so much weight on her shoulders, it is inevitable what has happened,” said Hamilton, who wrote in a recent social media post that he struggles with his mental health.
“She was incredibly brave and it is now about asking those in power to think about how they react because the way they reacted with the fine was not good.
“Someone talking about their personal mental health and then being fined for it was not cool.
“They could definitely have handled it better and I hope they take a deep dive into that and find a better way to navigate it in the future.”
Hamilton, 36, who walked out of a press conference on the eve of the Japanese Grand Prix in 2016 citing unfair media criticism, posted a message in support of Osaka earlier this week.
The seven-time world champion, who trails Red Bull rival Max Verstappen by four points ahead of Sunday’s Azerbaijan Grand Prix, admitted he too is uneasy with the media glare.
Hamilton was a year younger than Osaka when he burst on to the global sporting stage with McLaren in 2007.
“I have learnt the hard way, and made many a mistake,” he continued.
“It can be daunting standing in front of a camera. It is not the easiest thing to do, particularly if you are an introvert and you struggle under those kind of pressures.
“When I was young, I was thrown into the pit and I was not given any guidance or support. I was never prepared to be thrown in front of the camera or guided as to what to look out for and how to navigate through it.
“When youngsters are coming in, they are facing the same thing and I don’t know whether that is the best thing for them. We need to be supporting more because you shouldn’t be pressured.
“With Naomi’s scenario, she didn’t feel comfortable because of her own personal mental health and the backlash against her was ridiculous.
“People were not taking into account that she is a human being and she was saying, ‘I am not well enough to do this right now’. That needs to be looked into and how people react to that.”