This isnt so much an issue of could you
, as much as it is an issue of should you
Frankly, even if you were an expert marksmen, which, no offense, if you were you wouldn't be asking if .22's and deer go together anyways. Theres just too many factors here. A head-shot means 100% accuracy or you now have a severely wounded deer that will sadly probably live on for days. Missing ears, shattered jaws, large open head wounds, starvation and or death by dehydration.. The only thing those injuries have in common is a hunter who shouldn't be in the woods in my humble opinion.
The below video is pretty graphic, but illustrates why head shots even with higher cal rifles isn't always going to lead to a clean kill. YouTube - Headshots Aren't Always A Good Idea!