Depends what you consider rich. Most of them do pretty well. Pediatricians and GP’s generally are near the bottom of the salary list, @Caravanfan posted a good link. I know doctors who make “just” $90,000—$100,000 a year.
I think in most parts of the country someone making $200,000 plus a year accumulates wealth over time and by mid-life they probably are viewed as rich by most of America, but really it is an upper middle class salary. In very expensive cities $200k would definitely not be rich, because you spend so much on just living expenses. But, also think about it like this, there are CEO’s making $10 million a year!
Some doctors make a lot during some parts of their career and not others. My uncle was an ophthalmologist and he made a million dollars a year for about 10 years of his career, but when he started is was much much less, and towards the end of his career it was much less. Still a lot of money though.