Disagreements in human relationships are inevitable, even among those who are sincere believers. As Christians, we believe there are certain things which we should never even consider in times of difficulty. For example, when we are upset, we will never even consider killing our opponent. It's laughable to even think that's a possibility. In the same way, in marriage, we do not even consider divorce to be an option in resolving our disagreements. Divorce should be as unthinkable an option among spouses in marriage as murder is unthinkable among friends in an argument.
In addition to knowing what not to do, it's important to know what TO do in tough times. We should seek for tools to resolve our conflicts in a healthy, constructive manner.
In my own marriage, remembering the following two things has helped tremendously:
I must be the first to die to myself
There's almost never a circumstance where the blame is 100% on one party alone. Both parties own some portion of the responsibility for any disagreement. As the "head of the household," I believe the man should take it upon himself to lead by apologizing for his part first, even if he feels 99.9% of the blame is with his wife. (It's incredibly unlikely that this felt proportion reflects reality)
One thing I have learned is that I must not apologize hoping to trigger reciprocity. My goal should not be to get my wife to "apologize back," but rather to genuinely confess and own up to my responsibility. Sure, I may wish my wife would apologize, but that is a fleshly desire. I need to die to my will and my own desires, and seek instead to only be responsible for and concerned about my own shortcomings in the matter in question.
There's a lot of talk about the husband being a "spiritual leader" in the Christian world today. As I have often told newly wedded brothers, spiritual leadership is primarily comprised of being the first person in the relationship to die to yourself. There are all sorts of worldly ideas pervading Christendom about what it means for the man to be the head: about commanding respect, being obeyed, being the ruler of the home, etc. These are all wrong notions. To learn what true spiritual leadership is, we must look to Jesus Christ as our spiritual Head, and the Husband of His Church. Looking to our own Head and seeing how He spiritually led His Church, we see Jesus's spiritual leadership defined by denying His own will every day, dying to Himself, looking to His Father, depending on the Holy Spirit, carrying His cross, and coming up underneath us in service and love. He never demanded respect, or forced obedience, but rather set the example of humble submission and obedience to the will of God the Father.
This is what we as men in the home should do as well: set the example of humble submission to God as our spiritual act of leadership.
Seek restoration of my relationship with God first
One picture that has really helped me think about resolving conflicts/ disagreements in my own marriage is that of the hands playing a piano. These hands can be compared to the husband and wife in a marriage. Think of when a player's hands play beautifully. They are coordinated not through efforts of their own, time spent together in a kangaroo pocket, etc, but rather simply because they're both perfectly connected to the player's head.
In marriage, I used to think we needed to have lots of long talks to "get on the same page," etc, which really only amounted to the hands "spending more time together;" it never actually put us in tune! I mistakenly thought that unity was a function of coordination and communication, but more coordination and communication did not result in more unity; often, my own efforts only resulted in greater disunity.
When I saw that if the hands aren't coordinated, it means one of them must disconnected from the player's head, it all made sense! Disconnection is paralysis, and we can't expect a pianist with a paralyzed hand to play beautifully. Likewise, in marriage, we don't need to spend more time together as paralyzed members; we need to seek to be restored to perfect connection with our Head!
Seeking God individually first, examining my own heart (and asking Him to search my heart and reveal whether there's any offense there), eagerly resolved to make right anything He reveals to me, is the more productive and fruitful path to restoring my earthly relationship than belaboring various viewpoints and tactics in a discussion.
As we have done this, we have found many disagreements completely fade away entirely, needing no further discussion. When we do desire further conversation, we can engage fruitfully as members that have been restored to health as fully functioning, healthy hands.
Brother Zac Poonen has used the picture of the cross to describe all of our human relationships (the horizontal beam) in the context of our relationship with God (the longer vertical beam), and this picture is definitely true in marriage: no horizontal can thrive with a broken vertical; and almost all broken horizontals are really because of a broken vertical beam.
God's will for our marriages is that they would demonstrate the wonder of His redeeming love for us, reconciling us to perfect unity with Himself (Ephesians 5:31-32). These are a couple of ways in which we can personally seek to reflect His love in the disagreements that threaten to divide our marriages.