Important info missing
Gifts are often very personal and some spend a lot of time choosing them, they can also send messages.
How long has she been your girl friend? Are you serious? With my wife I could get away with it, but I would still be careful as it might be something special to her. As was posted, it is courteous to receive a gift graciously and until you are on very firm footing keep it. Do you try to buy gifts that have meaning to the other person? If so, you might be upset if they sold your gift. To me your girlfriend's position on her gift is very understandable, her position on a prize is not and is possibly a backlash from your reaction to her gift.
Good luck, if she is a terrific young woman please remember HT is stuff that will be obsolete very soon, terrifc young woemn have staying power. Worst case let this be a reason to have some tough conversations and understand each other. If all this sounds stupid, sell the watch, be prepared for a new girlfriend(s).
Good luck