WordNet records no synonym and no antonym for country doctor. It is the only member of its synset, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
a doctor who practices in the country (rather than in a city) usually remote from a modern hospital
“do country doctors still make house calls?”
Synonyms, antonyms and the broader and narrower relations for country doctor come from Princeton University’s WordNet. Full sources and licences.