Can my father's credit card bills be set aside?
To be clear, a person's credit card debt cannot be transferred to his or her family upon his or her death. This means that your father's credit card debt will die with him. I'm mentioning this because collection agencies will say almost anything to get the debt repaid, so it's important that you be able to distinguish what's true from what's not should you one day be the unlucky recipient of one of their harassing phone calls.
That said, there are a few things to consider in the here and now. Since your father has qualified for Medi-Cal I am assuming that he does not have any large assets. In this case, there is not much that the credit card companies can do. However, those with debt who do have assets such as a home could be sued by the credit card company and a lien could be placed on the property (possibly even force a sale) to pay back the debt. If your mother is alive and still living in a shared home for example, this would be something to consider.
Some credit card companies will forgive the debt if it's communicated that the person with the debt supports him or herself with Social Security alone (and in your father's case, even that is being taken by the nursing home). However, it would be wise to consult with an attorney before communicating with the credit card company to thoroughly explore the implications. In some cases the IRS may choose to tax your father on the amount forgiven.
While you're deciding how to proceed there is a way to stop the harassing phone calls. Under the Fair Debt Collections Practices Act your father can request in writing that the collectors no longer communicate with him about his debt. This may or may not work, but it is worth a try.