Most credit cards have a pay by phone option.

However, what the cc company does is a pull from your bank's checking or savings account. It would be really sweet if you could use a debit card to do the deed! (As an aside, what is really nice about pay by phone is that the CC company marks the payment at the time you do it. So if you forgot to make a payment, you have up to the day it is due to do it by phone without penalty )

Of course, you wouldn't want to use a CC to make the payment. That would be financially foolish, to say the least
Perhaps if you got a live agent on the phone, they might be able to do a end around using a GC but I doubt it.
Let us know if you are successful