On what grounds will the present-day Supreme Court uphold a law that treats women differently from men?
It is not always clear when the Supreme Court will or will not uphold a law that treats women and men differently. However, the general rule is that the Court will uphold such a law as long as the law has as its goal an "important" governmental interest and that the law is "substantially related" to the achievement of that goal. This is called "intermediate scrutiny" of "heightened scrutiny."
The Court has typically found that laws that treat men and women differently are much more acceptable than laws that discriminate on the basis of race or religion. In those cases, the laws as subject to "strict scrutiny" where the objectives have to be "compelling" as opposed to "important" and the law has to be "narrowly tailored" as opposed to "substantially related" to those goals.
In other words, the Supreme Court will strike a law down if it does not think that the law is closely connected to some important goal. Of course, this is a very fuzzy standard and so it is impossible to know exactly which laws will and will not pass muster.
Your question, of course, leaves room for a great degree of conjecture. The Supreme Court has long held (see Mueller vs. Oregon) that there are fundamental differences between women and men that render women less capable of performing certain work functions that men might perform without the same risk of injury. One arguing FOR the equal treatment of women might argue the Equal Protection Clause of the Fourteenth Amendment; however the present court tends to be very strict in its interpretation of the Constitution. My guess is, the court would hold that the Equal Protection Clause does not apply to gender differences; that this fact was recognized when the "Equal Rights Amendment" was proposed--that is, a Constitutional Amendment would be required to offer women the same rights. Since that Amendment failed, the right simply doesn't exist.
An analogy (albeit not the best) might be the slavery issue. Although many people argued against slavery, it was commonly believed by scholars of the day, including Abraham Lincoln, that it was constitutionally protected. This position was vindicated when slavery was ended only by the Thirteenth Amendment. The present court will probably take the position that dissimilar treatment of men and women is not violative of the Constitution, absent a specific Amendment to indicate as much.