The US doesn't have debtor's prison per se. Only certain kinds of obligations, which are already associated with "jailable" offenses can result in something that looks a lot like debtor's prison. In some cases not being able to pay a debt really should carry a jail sentence, for example in case of clearly fraudulent behavior.

John Oliver talks about a particular kind of "debtor's jail" in a recent episode of "Last Week Tonight". It seems though that appeal courts don't look favorably on such jail terms.