If my employer mistakenly withheld Virginia tax from my income, and I enter the amount, shouldn't that amount be deducted from my North Carolina tax where I live?

Sorry, no, it doesn't work like that. The VA taxes withheld were sent to the VA Dept of Revenue and they couldn't care less about your NC tax return.

You'll have to file a nonresident VA return, claiming zero income from VA sources, to get a full refund of the VA tax withheld.

To do that, run through Personal Info again and for the question about "Did you make money in another state?" answer Yes and select Virginia from the list of states in the dropdown. That will provoke TurboTax to prepare a nonresident VA return for you.

In your resident TT/NC return, do not go through the Step-by-Step for "Credit for taxes paid to other states" (or whatever it says about that) because you are not entitled to a credit for taxes paid to VA because your nonresident VA return has no tax liability on it (zero income).