I have an Apple gift card. I want to use it to pay my Apple Credit Card bill. Can I do that?

Apple Gift Cards are limited to the purchase of Apple products and services. Apple Card MasterCard is a co-branded credit card issued and serviced by Goldman Sachs Bank. These are two different companies and funds cannot be transferred between them.

Sorry, but no.

It can be used for AppleCare payments, but you cannot purchase AppleCare with it with the exception noted below.

What you can buy with your Apple Gift Card or Apple Account balance - Apple Support

The relevant portion here is:

For some recurring charges, like AppleCare+, you can't start your subscription with Apple Account balance. But as long as you keep another valid payment method on file and your Apple Account balance would cover an entire charge, your recurring charge will pull from your Apple Account balance.*

* In some countries and regions, your Apple Account balance can’t be used for some subscriptions like AppleCare+. Check if Apple Account balance in your country or region isn’t supported.

Yes, it’s a payment method for Apple Care. I’ve used Apple ID Account funds for monthly payments. See Jim’s comments above.