^^Title speaks for itself.
First, you should pay it off with fiat. After:
I would reverse the question, why not to use a bitcoin debit card if you have bitcoin and you want to use your bitcoin for spending? (paying card debt off is also spending purpose). Most of the cards are VISA branded so they are accepted worldwide. OK, bitcoin debit cards are not anonymous. But, if you pay off your card debt with bitcoin, tax authorities will surely ask about the origin of the amount you have paid off, so it also doesn't make any difference.
Every service does not accept Debit card. For some, Credit Card is mandatory, e.g. Meetup.com, Moniker.com, BestHostFree.com (though it accepts bitcoin directly) etc...
In this case, there are two options:
1. Convert the neccessary amount of bitcoin on an exchange and send it directly to your CreditCard's account nr. There could be a problem with this at the exchange, because your EUR, USD, etc account has your name, but the credit card account doesn't, it's the bank's account to keep the details of the credit. So it's possible that some exchange won't let you wire money to that account.
2. Convert then send it to your everyday EUR, USD, etc... account. Transfer it to your CreditCard's account, and voila... problem could be that in some EU countries, banks started to monitor if a wire transfer comes from a cryptocurrency related account number and sometimes they send the transaction back without a question. So test it, before you spend large amount from your credit card, and always be safe to be able to pay it back from another fund source. You would never know that your transaction will arrive or not, despite of the formers has arrived without problems, banks are arising on the topic it seems...
