Forgiveness is critical to resolving conflicts. If your spouse admits to a wrongdoing and apologizes, accept the apology, and if not, offer to forgive it nonetheless. If you plan to stay in the relationship, there’s no benefit whatsoever in holding onto anger. Of course, if the same problem keeps coming up, an apology is meaningless — actually, you have never really solved the problem, so it might not be an issue of communication style. However, barring repeated offenses, you will do a lot of good for your relationship by letting bygones be bygones.
On a final note, marital disputes are not about winning a contest or defeating your partner. They’re about maintaining and improving a relationship — together. When there’s a winner, there’s also a loser, and a loser doesn’t feel good from the experience. Truly effective communication not only leads to resolution and avoids escalation, but, in the end, each partner feels their point of view was heard and understood, they feel good about each other, and the relationship moves in a positive direction.
Ultimately, that’s the only outcome you should be looking for from disagreements.