Every government has always claimed the power to encourage good and punish evil among its citizens. The states that prohibited or otherwise regulated abortion were merely exercising this power. They concluded, on unassailable grounds, that abortion is wrong and they sought to limit it. The supposed right to privacy was always a red herring. No wrongful act is so private that it is beyond the reach of the ordinary government power to punish evil.
States punish spousal abuse and prostitution without giving privacy a second thought.
There never was any legitimate question about the constitutionality of abortion laws. Democrats could only sustain Roe by accepting a bizarre view of the judicial role. They had to portray judges as omnipotent to justify what the Court had done because it could not be justified in any other way. The left's academic wing rose to the occasion, producing reams of argument for the proposition that there is no theoretical limit to what judges can properly do to move society to the left. The more complex the arguments got the more they demonstrated their own emptiness.