"When you’re young, you’re thrown into the limelight and the spotlight and it weighs heavily on you. And probably most of us are not prepared," Hamilton told reporters at the Azerbaijan Grand Prix in Baku.
"I think she’s incredibly brave and I applaud her for her bravery because I think it’s now asking those in power, putting them in question and making them think about how they react."
Osaka was fined $15,000 for skipping a news conference following her first-round win and also warned of possible expulsion.
"The way they reacted was not good, with the fine. Someone talking about their personal mental health and then being fined for it, that was not cool. I think they could have definitely handled it better," Hamilton said.
Hamilton, who in 2016 refused to take questions from reporters during a Japanese Grand Prix news briefing, hoped organisers would find a better way to deal with such a situation in future.
"When they cry out for help, we should not be penalising them for those sorts of things. I know someone this year who has taken their life through mental health and it's deeply saddening an issue for so many people," he told Sky Sports television separately.
"We should be supporting people in those positions."
Recalling his early years, the 36-year-old told reporters he had "learnt the hard way" and made mistakes.
"When I was young, I was thrown into the pit and I wasn’t given any guidance or support," he said.
"What I do know is that when youngsters are coming in, they are facing the same thing as I did and I don’t necessarily know if that’s the best for them. I think we need to be supporting them more.
"I think it shouldn’t be a case where you are pressured."
Hamilton said the backlash to Osaka's initial refusal to do the news briefing for reasons of mental health was "ridiculous".
"People are not taking into account that she’s a human being and she’s saying ‘I’m not well enough to do this right now’. I think that needs to be really looked into and how people react to that," he said.
(Reporting by Alan Baldwin; Editing by Bernadette Baum) ((firstname.lastname@example.org; +442075427933;))