I had the same thing with my shed...6-8" space under.
I used 3/4 stone and shoved what I could under and then built the stone up to the bottom (with a rake)
So far no critters have tried to get under but I'm sure a determined one could. Not sure it the stone is a deterrent but seems to be working. It would be easy to see if something was going under as the stone would be dug out.
With screen I believe unless you put a barrier under ground that extends at least a foot beneath grade, anything would easily dig under it. Critters are a determined bunch when they want to get under something!
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