Privacy is the conclusion, not the premise
this decision ... stands for a broader principle: that government should not intrude on our most private family matters.
It's common to frame the right to abortion as a matter of privacy -- the government shouldn't be nosing in on women's decisions about their reproductive choices. A similar framing is used in opposing anti-sodomy laws -- the government shouldn't be peering into our bedrooms to see who we're having sex with. This is an appealing framing because it seems to settle the issue procedurally (via limits to the scope of legitimate government action) rather than having to engage in -- and hence hash out one correct answer to -- substantive questions like "is abortion wrong?" or "is sodomy wrong?" There's a tantalizing potential here for a securing broader agreement, so that people could say (like John Kerry) "I personally find abortion wrong, but since it's a private decision it's none of the government's business."
But I think resting our arguments on privacy is ultimately question-begging and gets the direction of inference backwards. What things are private cannot be defined in an objective, value-free manner. Rather, whether something is private depends on whether it's the type of potential wrongdoing that the government ought to be interfering with. Nobody can disagree that government should not interfere with private matters, because by definition "private" is just "that with which government shouldn't interfere."
Many progressive victories have come from arguing in this "wrong, therefore not private" manner. Most relevant in this context is the fact that spousal rape is now considered a crime, that is, a subject fit for government interference, rather than a "most private family matter" as it had once been deemed. Spousal rape wasn't turned into a crime by some sort of analysis revealing that it wasn't really private after all, and therefore it's subject to government control -- instead it was decided that it was a serious enough problem to merit government interference, and therefore it was ruled no longer private.
In my mind, the broader principle that encompasses abortion rights is the principle of defending the autonomy of sentient beings (such as women) and not according independent moral status to non-sentient beings (such as fetuses). We can establish who's sentient or not on the basis of external factual information, then deduce which situations do not implicate potential serious violations of sentient beings' autonomy (and ones in which one sentient party's autonomy rights would always trump any others), and then declare those situations off-limits to government interference. To say the choice to get an abortion is a "private" matter is to summarize or restate the conclusion, not to justify it.
Putting the "no government interference" and "private" ideas the right way around does not necessarily force the John Kerrys of the world to become pro-lifers. They could argue that the wrong doesn't rise to the level of seriousness to warrant government action, or that government action is counterproductive in this case, or that government action would have worse side-effects of various sorts. But they have to argue that, not just presume it by using the label "private."
*My enthusiasm for his statement is tempered by the fact that -- despite the gushing of various bloggy types about the spiffy new website -- I can't find this proclamation anywhere on WhiteHouse.gov.