I'm with Scott Lemieux on this one, Roe v. Wade is not the problem. Liberals, I think, are fairly desperate for some sort of rational explanation able to account for the conservative movement's all-consuming fury over this decision and their ability to channel it into a focused and highly effective strategy of judicial intimidation and lawmaker litmus tests. That Roe enraged them by being a major policy change (or codification, depending on how you look at it) instituted through judicial fiat makes sense to us because it plays into an outrage we could, conceivably, share.
Unfortunately, our security blanket has a bunch of holes. I doubt a single anti-choice warrior ever sat up and thought, "Abortion's one murderous practice, but what really galls me is the shoddy and undemocratic methods used to wedge it into the constitution." Rather, there's simply a surprisingly powerful belief that blastocysts are full-blown people and to abort one is murder. Moreover, this plays into a deep discomfort with the current social environment, gender equality, the sexual revolution, and a variety of other bits of cultural evolution that large swaths of the country would like reversed.
Now, there's a perfectly good argument to be made that Roe, in its way, hurt liberals by letting us relax over the issue. If we'd spent the past few decades massing pro-choice armadas and pursuing privacy laws with the sort of single-minded zeal usually associated with certain sorts of autism, we might have a stronger movement today. But that's neither here nor there, really, and the question is what to do tomorrow, not how things could've looked different yesterday.