After my Dad's passing recently while in the hospital, am I (as his daughter, and caregiver) responsible for his outstanding medical bills?

As long as you did not sign any document to be financially responsible, then NO.

Dad likely had Medicare as his primary health insurance, & Medicare is going to pay for the majority of the bills. Now if he had a good secondary policy or if he was also on Medicaid, then they usually will pay the difference. The issue usually ends up being that some providers are not on the secondary list of providers or that they do not take Medicaid. For those, they will send a bill to the deceased which would then become a debt against his estate.

Now if dad died and left any assets, then he has an estate. In theory - assuming Dad had a will and named you are executrix - you would go to probate to settle his estate. The debts will become a claim or lien against the estate. Just how they get paid and in what order is very much dependent on your state's death laws and how probate runs. If for some reason, the bills are huge, if you have your wits about you and your state enables you to run probate for a good period of time, you can negotiate the bills. Or hire a probate attorney to deal with this.

If dad has a home, that is an asset of the estate. If dad had a life insurance policy that had the estate as it's beneficiary, that too is an asset of the estate, etc. So did Dad die with assets?

Unless you guaranteed payment, you personally are not responsible, but if your father had sufficient funds to pay for them, and they are legitimate, it's appropriate to pay them.

However, when I called to cancel appointments previously made before my sister died, some of her medical providers said they would just write off the balances and close her account.

On the other hand, if your father left a will which designed someone else as Personal Representative, or if he had a trust which appointed someone else as Trustee, that individual is responsible for bill payment.

No house, no estate. He lived with us for the past several years. He had Medicare and decent secondary insurance, so I am hoping the balances will be minimal, and can all be settled with the (very) small amount which remains in the bank account. I do want to pay them if I can. Thx for the input!