I've been known to express sympathy with this view. Indeed, my very first column for the on-campus journal I write for forwarded this very idea (though more to spark debate--I wouldn't call myself invested in it). In this light, I'd like to point you to a spectacular debate on the topic "Should Liberals Stop Defending Roe?" The participants are Texas Law Professor Sanford Levinson (yes) and Yale Law Professor Jack Balkin (no). And while both make excellent points and are clearly extraordinarily intelligent, I've been nearly completely swayed by Professor Balkin.
There are, I think, at least four reasons why this is a bad idea.
First, one doesn't "give up" on constitutional rights unless one is already convinced that they aren't very important or don't actually exist. Should liberals have given up on Brown v. Board of Education in 1962 when the going got rough if they genuinely believed that racial equality was a fundamental right of human beings? Or to take an example near and dear to your heart, Sandy, should we have given up on constitutional limits on presidential power and constitutional prohibitions on torture because most Americans thought our repeated carping on these issues unpatriotic, and that was bad for Democrats? If we don't stand up for the constitutional rights we believe in when they are politically inconvenient, what is the point of having such rights? Thus, to convince me that we should give up on Roe you'll first have to convince me (and many other people, too) that the right to abortion isn't all that important to women's liberty and equality; or that despite its importance, Bork and Scalia were right and that there is no such right in the Constitution.
Second, we must consider the consequences. Although overruling Roe will not change the law of abortion in liberal states like New York, it will produce significant restrictions on abortion in a very large number of other states, and outright prohibitions in a handful of still other states. In a post-Roe world, abortion will probably still be available somewhere in the United States. Even so, we will probably return to a world (indeed, a world we are already approaching under current doctrine) in which abortions are freely available to the rich but not the poor. Obtaining an abortion in another state requires time to travel, making excuses (i.e., lying) to employers and to family members about one's whereabouts, and considerable expense. Many states currently have waiting periods, and no doubt more states will adopt them—with more draconian requirements-if Roe is overruled. Current waiting period requirements increase the costs of abortion considerably because they often require two separate trips. That expense-and the deterrent effect on the poor-can only increase in a post-Roe world. Lack of access to safe and affordable abortion for poor women increases health risks for those women, and condemns them to lives of increasing economic hardship and dependency, not to mention the costs to society as a whole. The Democratic party has long claimed to stand for sex equality and for economic justice. Capitulating on Roe is inconsistent with both commitments.
Third, the conventional wisdom that overruling Roe will simply return abortion to the states underestimates the strategy, the devotion, and the ambitions of the pro-life movement. If abortion is murder in Alabama, it is equally murder in New York. The pro-life movement will almost certainly push for a national solution to the abortion problem, which means that we may get more restrictive federal abortion legislation that will preempt liberal laws like those in New York. No doubt a nationwide ban on abortion is not politically feasible in the short run; what is feasible, however, even with the changed political climate that we both imagine, are significant restrictions on abortion at the federal level, especially if the Republicans maintain control over at least one branch of Congress. Moreover, if Republicans control the White House, they can do enormous mischief to abortion rights nationwide through administrative regulations that have the force of law and preempt more liberal state laws to the contrary.
Fourth, giving up on Roe in practice will take down more than Roe itself. It will put enormous pressure on other Supreme Court precedents that protect people from state interference in matters of family life, contraception, and sexual autonomy. The pressure is not logical but ideological. It is easy enough for a lawyer to distinguish Roe from earlier cases protecting the right to use contraceptives (Griswold, Eisenstadt, Carey) and later cases protecting the right to same-sex intimacies (Lawrence v. Texas). After all, neither contraception nor same sex sodomy involves the destruction of an embryo or fetus.
Nevertheless, this fails to account for how Roe would be overruled in practice. Imagine how one would "give up." You can't send secret signals to the liberal justices saying "psst, hey Ruth Bader Ginsburg, take a fall on the next abortion case." Rather, giving up on Roe means not opposing new Republican judicial nominees who are committed to overturning Roe (as opposed to merely limiting it). But those sorts of judges will likely oppose much of the other existing jurisprudence on sexual autonomy. The opinions they write will likely emphasize that it is wholly illegitimate for courts to discover and enforce rights not specifically enumerated in the Constitution (unless, of course, it's unenumerated rights that conservatives happen to like! See the federalism decisions). Whether or not cases like Lawrence are technically distinguishable by well-trained lawyers, they may not be distinguishable in the view of the new Supreme Court majority.
Balkin goes on to show off-topic impacts (two actually: the type of judge who will overturn Roe will also vote against liberals on non-sexual privacy cases, and abandoning Roe would make it part of the "anti-canon" of cases such as Plessy and Dred Scott--which will provide a foothold by which conservatives can push for massive changes in prevailing constitutional theory) and provide solvency (push for a Freedom of Choice Act to put Republicans on the record and legitimize Roe democratically). Would that my debaters could write like this!