It depends on the rules you were married under, which are determined by state law and your prenuptial agreement.
Whether one spouse is responsible for the other spouse's debts depends on the applicable matrimonial regime of their marriage, which can vary depending on where and when thay married, and on the prenuptial agreement (if any).
In the United States, the matrimonial regime is mostly determined by state law, so it depends on the state under whose laws they couple was married. In most states, spouse's property is legally kept separate (separation of property):
Common law is the dominant property system in the United States and has been adopted by 41 states.
The theory underlying common law is that each spouse is a separate individual with separate legal and property rights. [...]
IRS, Internal Revenue Manuals, Part 25, 220.127.116.11.1 - Common Law
However, nine states (Arizona, California, Idaho, Louisiana, New Mexico, Nevada, Texas, Washington and Wisconsin) and the U.S. Territories of Guam and Puerto Rico also have a community property system (which however does not necessarily apply to all marriages).
So, if separation of property applies to the marriage, one spouse is not liable for the other's debts (within the limitations described in ohwilleke's answer). If community property applies, then the spouse may be personally liable:
Community property includes all financial obligations (debts)
accumulated during your marriage or domestic partnership. This is true
even if the debt was incurred by only 1 of you, or even if a credit
card was in the name of 1 spouse or partner only.
Your spouse or partner may have gotten into debt in his or her own name that you are not aware of. If the debt was incurred during your marriage or domestic partnership, it belongs to you too.
California Courts, Property and Debt in a Divorce or Legal Separation, Community and Quasi-Community Property
However, even in that case there may be limits or special rules that apply the liability - so you'll have to talk to a specialised lawyer to be sure.