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

No, quite the opposite. If the spouse cheated then he/she did wrong and that shouldn't be overlooked. It's then entirely up to the wronged party to evaluate his/her marriage to see whether it's worth keeping AND/or to decide to what extent, if any they wish to accept the illegitimate children of their spouse.