Can an abused or abandoned spouse remarry?

No. Independent of the fact that motivates the separation or divorce, a second marriage is not permitted by the Lord.

Php 3:20-21

Now to the married I command, yet not I but the Lord: A wife is not to depart from her husband. But even if she does depart, let her remain unmarried or be reconciled to her husband. And a husband is not to divorce his wife.