"that there is no provision in the US Constitution that gives women the right to that kind of privacy."
What about men?
That is the far right's latest spin against Roe. While the Constitution may or may not provide a specific right to privacy, US law provides for doctor/patient confidentiality. That, my friend, is prudent law and provides a right to privacy.
If you think abortion started with Roe, you're kidding yourself. Prior to 1973, physicians performed D&C's in hospitals to terminate pregnancy for women who had the ability to pay. D&C is the same procedure used for spontaneous abortion and certain menstral irregularities. Wealthier women had access to safe, sterile abortion. Poor women got coat hangers, infection, and death.