If my father dies with credit card debt, as his daughter am I responsible to pay it

I am sorry for your loss.  Your father was responsible for his own credit card debt, and you are not responsible for it.  If he had no assets, you do not need to open an estate.  If the credit card companies call you, simply tell them you are his daughter, you were not on his credit cards, and you have no reason to open an estate for him.  If they contact you more than once, contact an attorney who practices in the area of fair debt collection.  Attempts to collect from you after you inform them of the circumstances could be a violation of the Fair Debt Collection Act.  It has significant penalties and allows complainants to collect attorney's fees.

So sorry for your loss.  If you are not on your father's credit card, you are not responsible for the debt.  Your father's estate is responsible for any debt, however, if there is no estate, credit card company will not be able to collect from you.  If they contact you, let them know that you are not on the credit card and your father left no estate.