The conservative response to egalitarian equality found in feminism is something called complementarianism. Feminism says that men and women are equal and similar, allowing for fluid interchange between men and women in the various areas of life. Conservatism rejects this and says men and women are equal and different, applying certain limits to men and women for the sake of roles geared specifically for each particular sex.
The problem with the complementarian idea is that it is still feminism, even as its advocates claim to oppose feminism. Its declaration that men and women are equal in value, but different in roles tries to emphasize the ideas of value and roles, but what is applied is equality and differences. The problem is made worse because progressives do not deny that men and women are different, only that those differences cannot be limited to specific roles. So the “equal but different” stance is the same for both the complementarian and the egalitarian. Both the conservative and the liberal are feminists.
In order to get away from feminism and its corrosive effect on culture, there needs to be another way to view men and women and that is to say that they share a common humanity. Both are human and humanity comes with its own inherent dignity. Men and women are not animals and should not be treated as such. Cruelty and maltreatment are rejected in favor of consideration. Hence, the Christian admonitions that a man should love his neighbor and a husband his wife.
However, this common humanity does not automatically mean equality. In society as a whole, inequality abounds and society is actually the better for it. When it comes to the health of a loved on, the doctor is over the murderer, though both are human. The man who tells the truth and keeps his word is over the liar and fraudulent, though both are men. Not all men are qualified to lead, to manage wealth, to fight for a cause though all men may be capable. Of course not. So, inequality is a part of life and it is not the evil it is made out to be.
In the same way, men and women share a common humanity, but that does not make them equal. Male headship is part of the natural order of life and female submission to men is part of that order. Women are simply not qualified for doing what men do simply because they are women. In the same way, men are not capable of doing those unique things women do because they are men. There should be no shame in this.
Men understand they are not women and find no shame in the fact that they lack those particular qualities that women naturally carry. However, modern women find it shameful that women are not doing what men do and push to leave behind their inherent femininity in order to pursue the world and work of men. At the same time, they push for men to find new-found shame over their lack of feminine qualities.
What drives the conservative / liberal push for equality is the idea that inequality automatically means devaluation. If every person is not equal to every other person, especially if women are not absolutely equal to men, then those who are not equal are less valuable. This makes sense if men and women are stripped of their common humanity and reduced to economic units, making money the measure of all things. If a man earns more money than a woman, then the woman is devalued, base solely on her potential for economic earning.
And thus we have the main issue behind the modern obsession over equality of the sexes. Men and women are measured by their ability to earn money for corporations and governments, under the guise of earning money for themselves. For the love of money, man is remade in the image of economic theories, forgetting that there is a God who made man for more than the pursuit of food and clothing. This make feminism and equality not just social theories or political rights, but moral evils and as such they should be opposed.
Let your manners be without covetousness, contented with such things as you have; for he hath said: I will not leave thee, neither will I forsake thee (Hebrews 13:5).
The face of the mortal sin of covetousness is equality.