It isn't wrong to look stuff up on the internet that is openly available to look at. It might not (as is being reported) something that makes the looker happy, but it is not legally or morally wrong to do so. Taking a judgmental unkind tone towards such commonplace activity is rarely useful in my opinion.
It is not stalking. Seriously - it just isn't. Look up the statute in your state (if in the US).
I was never envious of the woman I hired. I do not believe her life was better than mine and I have never known a therapist who was not screwed up in a lot of ways no matter the veneer. Many ( I believe most) of them become therapists because of how screwed up they were (bad childhoods, hate their parents etc) and the ones I have known personally (I have several in my wider circle of friends) have rotten screwed up children, have drug and alcohol issues in themselves or their families, are on their second or third marriages, and so on. They are not super humans.
Facebook in general seems to cause a lot of problems (I am not on it - just what I have observed) with people trying to appear different (better) than they are - projecting a false image.
I don't think it is uncommon (from reading what clients write) for a client to think the therapist leads a better life - whether one looks them up or bases it on other fantasy ideas.