The denial in the Arkansas case does mean something. It is at least mildly surprising, especially because the denial was not accompanied by any separate opinion from a justice dissenting or concurring in the denial.
But the denial most likely does not mean that the Court has done an about-face on abortion rights. That’s because the case, as decided by the courts below, actually did not decide whether the Arkansas statute is constitutional or not. The question presented, as put to the court by Planned Parenthood, was what level of evidence the reviewing court needed to decide the constitutional issue. The Eighth Circuit decided that the court did not yet have the evidence it needed; the case will go on until it does.
The important precedent in abortion cases is not Roe v. Wade (1973) but Planned Parenthood of Southeast Pennsylvania v. Casey (1991). In that case, a narrow plurality of the Court (Justices Sandra Day O’Connor, Anthony M. Kennedy, and David Souter) proclaimed they were reaffirming the “essential holding” of Roe. “Reaffirm” in this case meant “completely rewrite.” In place of the old Roe doctrine that the right to choose abortion was “fundamental” (like the right to free speech, or to marry or bear children), the plurality now proclaimed the “undue burden” test—a regulation of abortion before viability was valid unless it had the “purpose or effect” of blocking women’s autonomy to make the choice of abortion.
Over time, the doctrine has gotten fleshed out. A provision requiring married women to notify their husbands was no good—that could have the effect of giving the husband a veto. Waiting periods and abortion-negative “informed consent” materials, however, were okay. A federal restriction on so-called “partial birth” abortion was permissible, in part because women could still have legal abortions by other methods.
The most recent “undue burden” case, Whole Woman’s Health v. Hellerstedt (2016), is also probably the most important—and it was a win for abortion rights. Texas passed a massive “health regulation” statute. Courts allowed it to take effect—which promptly led to the closure of half the state’s reproductive-health clinics. But afterwards, a district court heard testimony and considered documents assembled by the parties. That court issued extensive findings of fact comparing the effect of the law on abortion availability (drastic) with the public-health benefits of the various restrictions on clinics (limited, perhaps nonexistent). It concluded the law was an “undue burden.”
The Supreme Court, 5–3, agreed and struck the law down. The Hellerstedt opinions are dense—Justice Stephen Breyer’s decision, for the majority, weighed in at 40 pages, almost all analysis of the facts found below; the principal dissent, by Justice Samuel Alito, was three pages longer and equally fact-intensive. The key to the Court’s decision was the majority’s conclusion that in this particular case, the law’s requirements provided “few, if any, health benefits for women, pose[d] a substantial obstacle to women seeking abortions, and constitute[d] an ‘undue burden’ on their constitutional right to do so.” Even Alito’s dissent did not argue for overruling Casey; instead he contested the majority’s interpretation of the facts.