Of course he's a public figure.
He doesn't just play golf for a living, take his winnings, and go home.
He puts his face out there in order to be more marketable. He makes MUCH more in endorsements than on tour, so his public persona is very important to his hefty wallet.
You could argue that his sports related endorsements are the result of his success within the game, but companies like Buick pay him for his image, not his putting.
This by no means is an attempt to justify the media circus surrounding his every bowel movement, but he is very much indeed a public figure.