Is a spouse morally obligated to accept children born outside his or her marriage due to his or her partner's infidelity?

Morally obligated - no. You are morally obligated not to kill the cheating spouse. Divorce is definitely an option. Or even possibly the idea of an 'open' marriage. It can be a sticky wicket if not divorcing, and the cheating spouse is active in the child's life. The tangled web some people weave....