In Minnesota, a spouse is liable for the medical and other necessary debts of their spouse, as long as they are living together at the time the debt is incurred, pursuant to Minn.Stat. § 519.05. This is regardless of whose name it is in. While hospitals will sometimes pursue a spouse, apparently even for end of life care, I have never seen a credit card company pursue spousal liability under this statute.
This article discusses what appears to be an advancing bill to repeal that provision.