I recently sent a payment to attempt to start paying on a medical debt. The collection agency returned it with a letter saying that it is an unacceptable partial payment that we must pay the balance in full or call them to make satisfactory payment arrangements. Isn't this refusal of payment to pay off debt?

You think you can get out if it on that basis? Sorry, but a creditor is within their rights to not accept a payment that is not agreed upon. There seems to be a myth circulating that as long as you pay something on a debt, that can forestall legal action. That's just a myth.