I couldn't get your link to work. This one, which doesn't have the last three digits, worked for me.
http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=us&vol=410&invol=113Without getting into the current political make-up of SCOTUS, the problem with Roe is that it revolves in part around the issue of viability of the fetus. That is, the Texas law in question was struck down in part because it made no distinctions other than "life saving" or not.
"3. State criminal abortion laws, like those involved here, that except from criminality only a life-saving procedure on the mother's behalf without regard to the stage of her pregnancy and other interests involved violate the Due Process Clause of the Fourteenth Amendment, which protects against state action the right to privacy, including a woman's qualified right to terminate her pregnancy. Though the State cannot override that right, it has legitimate interests in protecting both the pregnant woman's health and the potentiality of human life, each of which interests grows and reaches a "compelling" point at various stages of the woman's approach to term."Why this matters now is that the point of viability has changed. In 1973, the traditional trimester formula made sense. As medical technology advances, the timing of the point of viability decreases. Roe is often interpreted as a privacy case, and it is, but only to a point. This is an over-simplification, but according to decision in Roe, the privacy of the mother trumps any rights of "potential life" up to the point of viability. This suggests that the privacy right in question here is malleable.
Opponents of Roe argue their cases in various ways, but the most dangerous ones to the pro-choice point of view are those that target viability.
The other common attack strategy is an assault on the right of privacy itself. This has much larger implications, but it cannot be discussed without mentioning the dominant philosophy of the court. The right of privacy is a "living document" interpretation of the Constitution. Strict constructionists, which are soon to be in the majority, deny this line of interpretation is valid and may strike Roe on those grounds alone, impacting several other rulings in the process.