Comedian and actor Hart was taken to hospital after a vintage car he was a passenger in rolled down an embankment in the California hills.
Speaking on The Kelly Clarkson Show, Johnson said his pal was "a lucky man".
"Everything is good," he said. "I spoke with Kevin - I connected with him today."
He added: "And you know what? These things happen in life and thankfully he was strapped in nicely to his car seat."
Hart was supposed to be Clarkson's first-ever guest on her new TV talk show, so following the accident Johnson cut short his honeymoon (with his new wife's permission) to fill in.
The pair are close friends - having starred together in films including Jumanji and Central Intelligence - and the former professional wrestler, 47, confirmed he would soon pay Hart a hospital visit.
"I love the guy, he's one of my best friends," Johnson went on.
"And honestly, I mean, thank God, it could have been a lot worse. So, he's a lucky man, and he knows it, too.
"I wish him the best and a speedy recovery and I'm going to see him soon."
Hart and the car's driver were both taken to hospital following the crash in the hills above Malibu on 1 September.
The driver of the car was not under the influence of alcohol, according to a California Highway Patrol collision report.