Can your face go numb from a pinched nerve?

Yes: Facial and trigeminal cranial nerves can cause facial numbness but much less likely a cervical neck pinched nerve. Consideration of a stroke for face numbness suggests a full evaluation.

Pinched nerve: Facial sensation is provide by the 5th cranial nerve and is one that cannot be "pinched" such as in a cervical disc. If you have facial numbness then one has to evaulate for a stroke or a lesion around the 5th cranial nerve - that would require a consultation with a neurologist.