My husband and his sister are not close, though he is close to his brother. My sister and I really have not gotten along until recently - she is 40 and I'm 42. We have never been best of friends. If we were not related, we wouldn't spend time together as we have very little in common. I am there for my sis when she needs it, as he is for his, but we're not going to be busom pals.
Though the weight issue is the one that they have "vocally" with each other, they may have other reasons for not getting along. If your daughter's obesity is a result of bad eating habits, only she can make the decision to change, and I would think that she would need a doctor's help to do it. He can then be supportive within the guidance of a professional. If it's a medical and/or psychological problem, then he needs to get off her case or help, again within the guidance of a professional.
They need to be mature and polite. Don't ask for anything more. They don't have to like each other. They may never get along and you'll have to accept that.