No, medicine cats cannot have mates. This is because they will feel weird and are more likely to treat their kin over other clanmates.
Medicine cats can't ever have kits, tom or she-cat. This is because if they're always in the nursery, no cat would be able to heal the Clan. They'd be more likely to treat their kin then their other Clanmates, e.g.: If there was a battle and many cats were near death on the ground. If a med cat sees their daughter near death as well as another warrior-which would they save? A medicine cats MUST be unbiased when they treat their Clan.
Medicine cats are technically allowed a mate, but nobody sees it that way because taking a mate is almost the same as simply saying they were going to have kits. When the truth is they're only not allowed to have kits. LOOP HOLE: Erin Hunter didn't think that through apparently.