Certainly not impossible, but with this type of glass it's tough to get non removable scratches on them, especially without your knowing. So try something simple, just get a paper towel, bunch up one of the corners, get it a little wet, and try scrubbing it off with a good bit of pressure.
There's a compound called Rareox which is an oxidation of Titanium in powder form about the consistency of talcum powder and when mixed with a small amount of water turns into a paste. This is the same stuff used by professional glass finishers when polishing the cut edges of glass table tops. It takes quite a bit of rubbing with a cotton ball or fine cloth but will remove fine scratches and leave the glass crystal clear. I used this to removed scratches on my car's side window. Only thing is if the glass is Gorilla Glass it may not work or may damage the scratch resistance treatment.
Than you need a real polishing compound. If you work has a lab with cross sectioning or other types of polishing use some thing from there. Otherwise you need to find a hobby shop or buy some polishing compound for use with harder minerals than typical glass. The post above gave you one name of a TiO2 based compound.
I know how you feel. I don't have a screen protector any one (long story) and I've been paranoid like crazy. I have a diamond ring and THAT finger stays as far away from my phone as possible. I have my nails grown out and i am even contemplating cutting those off for the sake of my phone. I really need my invisheild..