In the USA, most debts can be resolved via bankruptcy (to some extent). The specifics of that are beyond the range of my expertise.
Federal debts are a different story. If you owe money to the government, and you default, they can garish your wages and file a lawsuit against you until the debt is resolved.
Since credit cards do not fall into federal debt (unless there's something I am missing), this would not fall into something that is "criminal". Something criminal would be fraud or identity theft.