No. The Code of Virginia classifies property as separate, marital, or a hybrid of both. Property obtained or debt incurred prior to marriage is included in the definition of Separate Property. The only way you could be responsible for the pre-marital debt of your spouse is if somehow that debt became co-mingled with marital debt and tracing it back out was impossible. This sometimes occurs with property, but is much harder to do with debts.
In finalizing a divorce, you want to be sure to resolve all marital debt, remove yourself from all joint accounting, and have in writing who is responsible for paying off specifics debts as part of your divorce decree (in a Separation Agreement or otherwise determined by a court).