In our union (Buenos Aires -- URBA) and, AFAIK, all other Argentine unions, there is a strictly enforced golden rule: "no doctor, no match". No match can start if there is not a doctor (and some standard emergency equipment, as a rigid stretcher, neck immobilizers) in the perimeter area. This rule applies to all levels of game, from juniors to elite seniors. It is the duty of the local club to provide the doctor, notwithstanding the right of the visitor team to have their own. If the 'doc' is not there, and has not signed (and stamped with his/her seal) the match card, your duty as a ref is to call the match over, and the local team loses by default. The ref must state this on the match card and leave the premises as soon as practicable: if you remain at the local club premises, and the game is played anyways despite the prohibition, you, the ref, could be found liable for any injuries and, in any case, will receive a (hard, really hard) punishment from the Society.
It is the doctor's duty to write and sign a note on the match card if any serious injury occurs, and the ref's duty to enforce that rule. Since there is not a good definition on how serious is "serious", our usual practice is to ask the doctor to write down the primary diagnosis for any injury forcing a player to leave the field of play for the rest of the match.
Having the doctor there is a great help for us, since we can always have professional advice if an injury occurs. You have always the power to overrule the doctor's professional opinion, but only on the "safe" side: you can't allow a player to keep playing on against medical advice.