Moore was rushed to hospital earlier this month to seek treatment for what was said to be exhaustion, and was released two days later.
"I think it's a stupid thing. I think it's stupid for anyone, whether they're celebrated or not. I don't believe their 911 call should be broadcast around the world, but that's my opinion," Clooney told The Insider about the call.
"What's happening and what people have to remember is that people are getting famous from Facebook and Twitter, so it's not just about people here, it's about everybody. There won't be any version of privacy. So it's going to be a tricky thing."
Clooney went on to say that his concern is also personal, adding: "It's always troubling when people you know and people who you care about end up going through a difficult time."
Director Patty Jenkins has said that Moore is feeling "awesome and strong" following her hospitalisation and is recovering well.
> Kirstie Alley: 'Demi Moore will get better'