What we do is...
To answer your question on refusal of service, I'll tell you what we do. We are a large Medical office with 23 physicians. What we counsel our front office staff to do is this, reschedule the patient so that they can have time to save up their co-pay or deductible. You have a very small chance that if you see them you will actually receive payment if you bill them. So re-schedule them. Its not refusing to see them, which can be tricky, its delaying their appointment. Okay, its a very techical difference, but since you have scheduled a future appointment.
Your doctor has provided the patient a service for which he deserves to be paid. When they are at the desk, ask to set them up on a payment plan for their previous balance. When you make the new appointment, make it clear to them they will be expected to make a payment on the old balance (at least once a month) as well as pay for the appointment on that day. They will either pay you, or move on to a practice who will see them for free!
Another important note, if they are urgently ill, you may want to route them to the nearest emergency room. And if in doubt, make the physician aware of the situation and ask him if he wants to see them. Its ultimately his call.
I hope this helps.