That is ironic. The Roe vs. Wade decision was based on the Fourteenth Amendment, which ended slavery.
Yes, but the decision is more complex than that. Basically they say that when life begins (in the sense of this discussion -- in some sense obviously even the sperm are alive) is a philosophical and theological issue on which the court shouldn't take a position.
They review the history. Until the 19th Cent abortion wasn't always illegal. There was little sign in the common law of it being considered murder. Current laws started in the late 19th Cent. While there were probably other motivations, during that period abortion was still dangerous. Hence the Court considered the 19th Cent laws justified on medical grounds.
However by the time they made their decision, it was no longer dangerous. Hence there was no remaining medical justification, and they didn't consider other justifications constitutional.
Their decision that it should be allowed without restriction in the first trimester had nothing to do with viability, as I had always assumed. Rather, they believed that until the end of the first trimester, it was less risky to abort than to continue the pregnancy. That means there was no medical reason to restrict abortion then. (Incidentally, the mortality rate of abortion has continued to go down, while the mortality rate of pregnancy has actually been going up, so at this point their reasoning might lead to a later point in pregnancy.)
While the term privacy is used, I think that's misleading. Rather, the concept is that the government has no right to restrict citizens where there isn't good reason. The fact that a majority of other citizens have philosophical or religious objections is not a good reason. Remember, rights are not limited to the Bill of Rights. Indeed a number of the Founders objected to the Bill of Rights because they worried that it would cause courts to enforce only rights explicitly mentioned there. There's a more basic principle that the government can only limit citizens where it serves a legitimate government purpose.
They applied this principle using the "due process" clause of the 14th Amendment, but it's a basic principle behind the American Government. Other courts had used the 9th Amendment. I don't think any amendment is actually needed.
Part of the reason I go through this is because a lot of people say "there is no right to privacy in the Constitution. The Court just made it up." But there very definitely is the idea of government limited to specific purposes, and that's the principle they really used. If you're going to attack the decision you need to do it differently. If we lose the concept that the majority can't impose its philosophical views on a minority, we all lose a lot, but in the long run conservatives are probably most at risk.