Hard water stains are basically mineral deposits, so any kind of acid should work. To keep contact instead of the acid running down the door, you can try using it and leveling it outside or something. Make sure to protect the door and hardware though.
Depending on the water and other stuff, you might also have glass etching instead of stains. That's when silicates in the way have combined with the minerals in a way that pits the glass. If that's the case, you'd have to polish it out.