For the most part, the fight against domestic violence has been lead by left-leaning people. I am making a case that conservatives should take a strong stance against domestic violence as well.
The reason is not gender equality, and it is not nonviolence. Conservatism believes in neither. Rather the reasons are two concepts central to conservatism: Character and incentives.
Much conservative ideation revolves around the concept of character. A man who beats up on people who can't defend themselves from him does not have character; he is a coward and a slimeball. One major part of character is self-control, and another major part of character is refusing to abuse one's power. From the perspective of character, it stands to reason that violence toward women is incompatible with character; which means that it is incompatible with conservatism.
Another concept central to conservatism is that of incentives. The conservative wants to make profitable the actions that are beneficial and not profitable the actions that aren't. There needs to be an incentive on men to act with character; and that means, to avoid being violent to their wives. For this reason both the institution of divorce, which makes it possible for women and children to leave situations of violence, and the institution of jailing men who are guilty of serious violence, work for the benefit of conservatism. They make it profitable for men to behave with character toward their wives and their children and not profitable for men to act without character toward their wives and their children.
Thus, according to conservatism itself, there need to be strong laws in place - and strong enforcement of these laws in place - to prevent violence toward women and children. Men need to be taught to behave like gentlemen, and strong measures need to be taken against men who do not. Character most certainly demands controlling oneself around the people who are closest to oneself. And incentives, whether the right for the woman to leave a man who practices violence toward the woman and the children, or criminalization of domestic violence, must be in place to demand this gentlemanly behavior.
The fight against violence to women should not only involve people on the Left. The conservatives should be in the front lines of this fight as well, as doing so is a natural outgrowth of conservative values. Character most certainly means controlling oneself around the people closest to oneself. The men of goodwill will do that anyway. But for men of ill will, there is a need for incentives to act with character.
A lot of men who practice real abuses toward wives and children hide behind social conservatism and then go around posturing that they have character or values and that other people do not. They don't have character, and they don't have values; and for as long as these people use conservatism in that way, that discredits conservatism and its claims of supporting family values.
If there really is to be a pressure on people to behave in more righteous ways, then that most certainly means a pressure to end violence against women. A man with character would never allow himself to behave in that way; and men who would must be subjected to strong incentives against doing that. If conservatism is serious about its claims of possessing character, then it will fight violence against women. And if it does not do that, then it cannot be allowed to maintain its claim of being in favor of character and family values.