I would have to say until he's hung up his clubs, Tiger Woods will be the "face" of golf. There are people out there who have no idea who Jack Nicklaus is and only know who Arnold Palmer is because of the drink. But everyone knows who Tiger Woods is for whatever the reason; his golf accomplishments, the fact that he's black, the fact that he cheated on his wife multiple times, the popular opinion that he's an overall jerk, or the belief that he will never win another Major.
However, I think if anyone can overtake Tiger Woods as the "face" of golf, it will be Rory. He's wholesome, he's a gentleman, and he's a damn good golfer. He may never have the profile that Tiger has, I personally think that Tiger Woods is a once-in-a-generation talent, but I think he can be a guy who people will be able to recognize.
My dad says it could be Phil Mickelson, but I think it's too late for Mickelson to become the "face" of golf. If he would have won some of those Majors back when he was coming up short to Tiger year after year, then I would think it's possible.