In the event that someone is injured on my property and they ask us to file a claim to cover their medical insurance co-payments, would the injured party's medical insurance refuse to make payment?  Would my homeowners insurance be liable for the full medical payments vs. just the co-payments? 

Answer:

I have heard of people filing claims to cover their co-payments; however, the problem is that the original medical insurance may then deem that they are not responsible for any of the cost. So, it is possible that the injured party's medical insurance would then refuse to pay, and would push the full cost of any medical treatments to the homeowners insurance company.

If your homeowners insurance is deemed to be liable, it would be for the full medical payments in this case, and not just the co-payments.