There are serious concerns regarding the safety and health hazards involved in body piercing, especially certain areas of the ear and tongue. If your daughter agrees to have her navel pierced under the hygienic conditions insisted upon by her doctor, and she understands what she must do to prevent that area from becoming infected, you should give her permission to pierce her navel.
Allowing her to pierce her navel does not mean that you are a permissive parent or that you gave in -- or will give in -- in the future because she stopped talking to you. It means that you gave careful consideration to a wish of hers and agreed to grant it after much thought and discussion with her regarding her responsibility in the matter. If she doesn't agree to have her navel pierced under safe medical conditions, I would refuse to allow it to be done.