You can not target or kill the white rabbit.
From the fortune teller, head east along the riverbank (on top, not down at the water); you should spot it before you reach the rocks. It will then lead you north and then a little west. It should wait for you if you fall behind (though may still be hard to see, or obscured by foliage, depending on where you are standing), and hop away in a cutscene once you get close enough.
