Who is responsible to pay for my Fathers credit card debt after he passes when his estate isn't enough to pay the bill.

If the beneficiaries of the estate (heirs) take any assets that are left, creditors might be able to claim access to those assets for payment of the debts. With the credit card, if he was married and used the card to purchase family items, the credit card company could come after the surviving spouse (due to the family expense doctrine). If there is no surviving spouse, and if his assets have been exhausted to pay for his medical care and other expenses, then the answer may be - no one. If there is nothing in his estate, there are no assets to pay the creditors. Someone will need to obtain death certificates and notify the creditors of his death and of the fact that there are no assets. Creditors who believe there may be assets anyway have a right to open a probate estate to attempt to collect the debt; however, there will be nothing to collect if there is, indeed, nothing in the estate.