The right to privacy came from the Griswold decision and Roe used that same right to privacy to say abortion should be a decision between woman and doctor.
Any law prohibiting abortion would involve looking into whether a woman had an abortion and violate her privacy, but criminal laws are allowed to do that in the U. S.
The real problem with the new Texas abortion law is that a state can deny any constitutional rights by allowing citizens to sue somebody for exercising those rights. A gun dealer could be sued for selling a gun, a person could be sued for exercising his free speech about some subject, a person could be sued for exercising his right to an attorney, remaining silent, etc.
If a civil lawsuit can be used to enforce the abortion law it could be used to enforce restrictions on any rights.