Can I fry an egg in water?
You can fry an egg on a non-stick pan without any oil or butter, however you will get a better result with some oil or butter. The reason you use oil or butter is to:


Seal the pores of the pan
Provide quick heat transfer from the pan to the egg


Water doesn't do either for you, all you'll do is cool off the pan. Your egg will still cook, it just won't add any benefit. 

You don't need much oil, if you have greasy eggs you are simply using too much. A small amount of cooking spray is enough, you can also wipe a bit of oil on the cold pan.