You say that you argue over stupid things, is the relationship important enough to you both to cease and desist the arguments? You are very young and so your relationship may not last, there is still so much growing up to do. If your're in a relationship where there is constant fighting, cut the ties that bind you and find someone more agreeable.
I agree with the doctor that an eating disorder should be managed, with therapy. You need to keep your body healthy so that you don't get avoidable diseases later in life, like cancer. Look at the pictures on the websites that show you what may happen should you not get highly responsible with your nutrition. I follow Dr Joel Fuhrman's diet and have reversed diabetes, high blood pressure, high and low cholesterol. Nutrition means everything to your health.
You can make many friends here, on the adult or teen forum, and i hope that you find many supports here. You've come to the right place, to no feel alone during this time. God Speed.
You should probably seek weekly counseling. It sounds pretty serious if you are having the urge to control your eating in such extreme ways.