what sized screw should I use?
If you have doubts about the ability to solidly attache the screen with two screws into the wall, I'd consider putting up a 1x2 or 2x2 strip, then attaching the screen to that.

You should be able to use a number of screws to attache the strip to the wall, and if you're not concerned about removing it you could even glue it.  Then you can screw the screen to the strip, which should hold it without any problems.

If you need a completely flat mount for the screen, you can build a frame out of the same material.