A government ad urging young men to register for the draft packs a lot of stereotypes into 30 seconds. A mother, drying dishes in her kitchen, nags her son, who has just turned 18, to “just do it now before you forget.”

Her son, skinny and squeaky, pulls out his phone, registers and is transformed into a buff, deep-voiced adult. “Johnny!” the mother exclaims, while his little sister looks on in amazement. “You’re a man!”

Since 2016, women have been allowed to serve in every role in the military, including ground combat. Unlike men, though, they are not required to register with the Selective Service System, the government agency that produced the ad and maintains a database of Americans who would be eligible for the draft were it reinstated.

Almost all other sex-based distinctions in federal law have been eliminated, in no small part because of the pioneering work of Ruth Bader Ginsburg, who litigated women’s rights cases in the Supreme Court before she joined it. But the requirement that only men must register for the draft remains.

The court will soon decide whether to hear a challenge to the requirement, in language that could have been drafted by Justice Ginsburg when she ran the Women’s Rights Project at the American Civil Liberties Union.

“It imposes selective burdens on men, reinforces the notion that women are not full and equal citizens, and perpetuates stereotypes about men’s and women’s capabilities,” lawyers with the A.C.L.U. wrote in a petition on behalf of two men who were required to register and the National Coalition for Men.

Justice Ginsburg, who died in September, argued six cases in the Supreme Court. The first, Frontiero v. Richardson in 1973, also concerned the military. She persuaded the court that the Air Force’s unequal treatment of the husbands of female officers, who were denied housing and medical benefits that the wives of male officers received automatically, violated equal protection principles.