Such a requirement would go against Natural Law.
The Founding Fathers of our Constitution knew what Natural Law was. It is the natural knowledge, which all people have from God, of what is morally right and wrong. Sending women into battle - and making it possible for them to be drafted to do so - goes against what is morally right. Women are special, unique creatures, who should be protected—not sent into battle. Women have the incredible ability to create children; we should not send that which creates and houses human life, into battle, to be killed—neither should we make it possible to do so.
Men and women have different roles in society. The general role of woman is not to be drafted to fight in battle.
Men and women are different. They have different skills, traits, and abilities; yet these differences complement each other, and work in harmony. Simply from natural reason, we see one of the roles of men is to be the fighters and protectors. Basic anatomy makes this clear. Anatomy also makes clear what the role of woman is (again, I am speaking in general terms—I understand there may be exceptions); i.e., to bear children. This is a wonderful gift, which should be cherished and encouraged. Requiring women to sign up for Selective Service does the opposite than cherish and praise what makes women unique (bearing children being only one of the many, which are too many to list).