Sorry if I wasn't clear. Gun stocks can be a variety of different woods, though walnut is pretty much the most common. But various manufacturers have used all kinds of different finishes -- from stains and dyes to color the wood itself, to various surface finishes -- varnishes, polyurethane blends and such -- to simple linseed oil rubbed in well in many applications.
Remington has used a variety of finishes as well. My BDL had a super high-gloss finish (like an old Weatherby) that I kind of hated because it was shiny and also got damaged so easily. Most of the time, top coat applications like glossy polyurethane are very hard to fix.
If the damage is a DENT, where the wood fibers are compressed but not sheared through, steaming the surface with an iron can get those to "pop" back up to a degree. If the fibers are cut (and when you say "scratch" that's what I think you mean) then steaming won't do a lot of good, but it's easy and you might as well try.
To make them really disappear, you'll probably need to strip the stock and refinish it. Before going to that extreme, try the simpler stuff first.
You may be happy enough, if you simply apply a bunch of coats of dark brown paste wax and buff it out. You can build at least a little bit of finish back that way and often make the scratches unnoticeable enough to live with.
If you've really scratched all the way down to bare wood that looks lighter than the rest of the stock, you may want to start with stain pens. You can buy them at Home Depot or Lowes and they're like an old-school permanent market but full of (usually Minwax) wood stain. You won't match the color exactly, but that doesn't matter so much. You really only need to darken it a bit (go with lighter rather than darker, here) and get the shade of brown/red close. Then wax it and the scars will be about as unnoticeable as you're going to get.
Good luck!
-Sam