On privacy grounds? Ain't gonna happen.
Roe's main premise is that the unborn are not persons vis a vis the Fourteenth Amendment. Even Blackmun admitted that if the unborn were persons, the Fourteenth applied.
From Roe:
"The appellee and certain amici argue that the fetus is a "person" within the language and meaning of the Fourteenth Amendment. In support of this, they outline at length and in detail the well-known facts of fetal development. If this suggestion of personhood is established, the appellant's case, of course, collapses, for the fetus' right to life would then be guaranteed specifically by the Amendment."
But, in response to the clear words of the Fourteenth Amendment...
["No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws."]
They found this:
"All this, together with our observation, supra, that throughout the major portion of the 19th century prevailing legal abortion practices were far freer than they are today, persuades us that the word "person," as used in the Fourteenth Amendment, does not include the unborn."
Privacy was simply the excuse.
The right to privacy is a very real one, and has been recognized as "the right to be left alone," all the way back to the Marshall court.
"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized." - the Fourth Amendment
In its broader sense, though, it is unenumerated, as are many rights. That's why we have the Ninth Amendment - to preclude the violation of rights just because they're not listed. [For example: You have a right of privacy that assures that you can raise your children pretty much the way you see fit. That is not listed in the Constitution, but the right is an obvious natural right that few are dumb enough not to see as self-evident.]
But, they made this largely unenumerated right, privacy, trump a God-given right that IS enumerated repeatedly in our Constitution, in the Preamble and the Fifth and Fourteenth Amendments, and in prior founding documents such as the Declaration of Independence, the God-given, unalienable right to live.
IOW, they misused the right to privacy. That doesn't mean it doesn't exist.
And, even if by sheer political force you did overturn Roe on the basis of privacy, we would be worse off than before in many ways. The personhood of the unborn would still be unprotected and unestablished, and the very real right to privacy would have been declared nonexistent.
I don't see overturning Roe regarding privacy. I instead see it as reversing an activist reading of the 9th via Griswold, which turns the core meaning of the 9th on its ear.
I agree there is a right to privacy. However, it is not a federal perogative to enforce that right. The 9th simply tells the feds they have no say in the matter, other than aspects of privacy addressed in the 4th and 5th.