I posted about my husband going to rehab on Facebook
Will posting about his private matters cause problems for him?
Maybe but we are broken up right now.
If the intent is to demean, propagate lies or half truths about people, or designed to hurt, denigrate and destroy reputations behind people's backs, then spreading this news has crossed a line into harassment. Keep your private maters off of Facebook and try to work through things with your husband in peace.
He did it to himself, I can post about whatever I want.
Posting these kinds of things reflect poorly on you too. Even if you are separated you should still try to act civil and hope he gets better from his addictions.