As a generality the construction "He wipes X on Y" suggests that X is the active noun and Y is passive (being acted upon), whereas "He wipes X with Y" suggests that X is passive and Y is active. So if the napkin is being moved to wipe the mouth then "wipes his mouth with a napkin" is correct, but if the mouth is being moved to the napkin then "wipes his mouth on the napkin" is correct.
As you can imagine there are a lot of situations where there is a clear distinction of which is right such as "He wiped his shoes on the doormat". The other option would be "He wiped his shoes with the doormat", which would imply that he picked up the doormat and used it to wipe his shoes", which is pretty unlikely.
However there are obviously many cases that can go either way, and it is up to the writer or speaker to pick which is more appropriate.
In the case you bring up it is definitely more common to say "He wiped his face with the napkin", but Raymond Carver may have picked to use the other construction explicitly because it is unusual and it would imply different things about the person or the action being taken.