Golf is a game. Much like bowling is a game. Neither are sports.
I guess I would need to understand your definition of a sport.
Mine is an endeavor where at least two contestants compete under a uniform set of rules and conditions; where the outcome is based on the skill of the contestants; and where the contest includes the use of strategy to overcome the opponent.
I think based on my assessment, bowling is NOT a sport, ice dancing is NOT a sport, Cheerleading is not a sport, but having been stymied on the green by an opponent I can surely attest that golf is a sport.