On this forum, we have to go with just the information that you provide. Any other facts that you have might end up changing the answer.
However, on just the facts that you have given, as long as it's just his name on the card and that's it, it would be the responsibility of his estate to pay the card. If you are the beneficiary of his estate, you will need to deal with those debts. If you don't feel as though you want to handle it, retaining an attorney to be your representative for that transaction might be a (relatively) inexpensive way to get them out of your hair.
Whether you, as a spouse, is liable for your deceased spouse's credit card debt likely depends on what the money was used for.
RCW 26.16.205 (Liability for family support – Support obligation of stepparent) provides: "The expenses of the family and the education of the children, including stepchildren, are chargeable upon the property of both spouses or both domestic partners, or either of them, and they may be sued jointly or separately."
If the money was used for the sole benefit of the deceased spouse (for example, for gambling), you likely would not have any liability. On the other hand, if the money was used to buy groceries for the family, you can be found to be liable. The creditor would need a court order to make you personally liable.
If your spouse died solvent, his estate may have assets which must be used to pay valid claims that are timely filed by creditors.
Depending on how much money is involved, the creditor may make a lot of noises demanding payment but do not take the necessary legal steps to get court orders.
You may want to review your facts with an attorney, especially if you have minor children who may be entitled to Social Security benefits based on their father's earning history or other claims.