To follow up on Adam's point here, the conservative project to separate the implied constitutional "right to privacy" from Roe v. Wade is longstanding. (I should note, contra to Ponnuru's fancy shuffling, the question -- and Palin's answer -- were both about the "right to privacy" as opposed to whether specific provisions of the Bill of RIghts protect "privacy." Although I guess it's good to see a conservative admitting that Douglas' "penumbras and emanations" argument is, in fact, perfectly logical!) Reagan's solicitor general Charles Fried, for example, has argued that the Court could "pull the thread" of Roe without affecting Griswold and the general "right to privacy."
The problem with the argument that privacy could entail a right to use contraception but not a right to an abortion is that it's absurd. As Stevens memorably pointed out in Thornburgh:
For reasons that are not entirely clear, however, JUSTICE WHITE abruptly announces that the interest in "liberty" that is implicated by a decision not to bear a child that is made a few days after conception is less fundamental than a comparable decision made before conception. There may, of course, be a significant difference in the strength of the countervailing state interest, but I fail to see how a decision on childbearing becomes less important the day after conception than the day before. Indeed, if one decision is more "fundamental" to the individual's freedom than the other, surely it is the postconception decision that is the more serious.
If there is a fundamental right to use contraception, there must be a fundamental right to choose an abortion, and given how abortion laws are actually written and enforced it's nearly impossible to argue that a state's interest in protecting nonviable fetal life could trump a woman's fundamental reproductive rights.
--Scott Lemieux