Good question.
The principle of male-female (and racial, etc.) equality under the law is known as the "Equal Protection Clause," and it's found in the 14th Amendment.
However, men and women are obviously biologically different when it comes to reproduction. Men can't get pregnant, and therefore they don't need abortion rights.
So when it comes to the analysis of abortion rights, we have to look somewhere other than the Equal Protection Clause. The Supreme Court in 1973 found that the Due Process Clause, also found in the 14th Amendment, provides a "right to privacy" applicable in this instance:
"In January 1973, the Supreme Court issued a 7–2 decision holding that the Due Process Clause of the Fourteenth Amendment to the U.S. Constitution provides a 'right to privacy' that protects a pregnant woman's right to choose whether or not to have an abortion. However, it held that this right is not absolute, and must be balanced against the government's interests in protecting women's health and protecting prenatal life. The Court resolved this balancing test by tying state regulation of abortion to the three trimesters of pregnancy: during the first trimester, governments could not prohibit abortions at all; during the second trimester, governments could require reasonable health regulations; during the third trimester, abortions could be prohibited entirely so long as the laws contained exceptions for cases when they were necessary to save the life or health of the mother. The Court classified the right to choose to have an abortion as 'fundamental', which required courts to evaluate challenged abortion laws under the 'strict scrutiny' standard, the highest level of judicial review in the United States."
https://en.wikipedia.org/wiki/Roe_v._Wade
Every abortion-related decision since Roe v. Wade -- and there have been quite a few -- has accepted this fundamental analysis. To my knowledge, there's not a single justice, liberal or conservative, on the current Supreme Court who disagrees with it.
For me, when it comes to abortion rights, the "right to privacy" is about recognizing that the government cannot decide a woman's destiny whether or not to choose to bear children. At least, up to a certain point.
But in the third trimester, the Roe v. Wade court -- and I -- would agree that states can regulate abortion in the interests of preserving fetal life.