The US armed forces have used a draft of some sort in the American Revolutionary War, the American Civil War, World War I, World War II, the Korean War, and the Vietnam War. This draft has always fallen on men only. Since the end of the Vietnam War, men have been required to register or a possible future draft. There have been challanges to this as a violation of the Equal Protection clause because registration is only required of men, but none have succeeded as of July 2021
In National Coalition for Men v. Selective Service System No. 19-20272 (5th Cir. 2020-08-13) The 5th Circuit court off appeals considered a challenge to the provisions of current Selective Service act (50 U.S.C. §§ 3802(a), 3809) that require male US citizens of ages 18-26 to register for a possible future draft, but not women. The Fifth Circiut wrote:
Because that judgment [the district court decision holding the law unconstitutional] directly contradicts the Supreme Court’s holding in Rostker v. Goldberg, 453 U.S. 57, 78–79 (1981), and only the Supreme Court may revise its precedent, we REVERSE.
In Rostker, the Supreme Court held that the male-only Selective Service registration requirement did not offend due process. 453 U.S. at 78– 79. The Court relied heavily on legislative history showing that Congress thoroughly considered whether to require women to register. ... Women were then barred from combat, so the Court examined the constitutional claim with those “combat restrictions firmly in mind.” Id. at 77. The Court concluded:
This is not a case of Congress arbitrarily choosing to burden one of two similarly situated groups. . . . Men and women, because of the combat restrictions on women, are simply not similarly situated for purposes of a draft or registration for a draft.
The fifth Circuit opinion went on to cite State Oil Co. v. Khan, 522 U.S. 3, 22 (1997) to the effect that a lower federal court may not overrule a previous Supreme Court decision, even when the facts have changed, undermining the reasoning in the prior decision.
The Plaintiff petitioned the Supreme Court to hear the case and overrule Rostker, but it declined, in significant part because Congress is now in the process of deciding whether to alter or abolish the Selective Service System in light of women now being allowed to serve in all combat positions, and other changes since the Selective Service law was passed, a Congressional commission having recommended that registration be required for both men and women.
The history is discussed in the Wikipedia article on Rostker v. Goldberg and in the article on Conscription in the United States. The article on "National Coalition for Men v. Selective Service System" is also relevant.