How can I repair my wall where the paint around screw holes has torn away?
Looks like paint or a light texture on drywall to me, but I agree with DMoore otherwise. You'll want to take a couple steps:

Using a sharp utility knife, slice away the loose edge of the torn paint. Remove anything that isn't well attached, but don't cut through the paper on the face of the drywall if you can help it.
Using a 6&quot; or larger putty/drywall knife, skim the damage with general purpose or topping joint compound. (Spackle is usually very light and a little fragile for this purpose.) Press firmly with the knife laid over at a shallow angle. Don't let more material on the wall than it takes to fill the very thin void left by the removed paint/texture. Let that dry completely. A fan will make it quick.
Using a foam sanding block or a drywall screen, very lightly sand the repair flat. Use long strokes with light pressure. Your main goal is to eliminate the line around the edge of your patch where it blends into the paint. Pay particular attention to that as it's the most frequent cause for disappointment in repairs like this.
Repeat steps 1 and 2 if needed to get a perfectly smooth repair.
Using a heavy nap roller (say 3/8&quot; or 1/2&quot;), roll drywall primer over the repair. Blend outward with a relatively dry roller.
Do the same with the paint topcoat (which may need to be applied to the entire wall section for color matching). Use plenty of paint and don't overwork it. You want the nap to leave a heavy stipple. This should be adequate in lieu of sprayed texture, assuming the wall ever had that.

If you find that the texture just isn't heavy enough to blend you may need to pick up a can of &quot;orange peel&quot; spray texture. Apply that between steps 4 and 5. Practice on a scrap of cardboard to get a feel for distance.