The lynchpin to the Republican anti-transgender crusade is President Trump's January 25, 2025 Executive Order entitled "Defending Women from Gender Ideology Extremism and Restoring Biological Truth to the Federal Government."
This EO puts forth three critical "arguments" (better characterized as "assertions") that form the foundation for the enforcement of the order and all other EOs regarding transgender American (not competing in athletics, etc.)
It is important to understand that most of this Executive Order is either window dressing (an opinion section at the beginning about, grunt, grunt, scratch, scratch, "transgender bad") or are specific directives to the State Department regarding transgender and nonbinary documents. The first opinion section is legally meaningless in all senses except propaganda. The directives section only has force of law if (a) it falls within the scope of the President's power to administer the government AND the legal assertions underpinning the directives are sound (or at least make some kind of approach toward being rational).
In this case the President would probably win the argument that he has the requisite authority over the State Department issuance of passports, since former President Biden permitted the change in gender markers on such documents by Executive Order in 2022.
On the other hand, the legal representations in the body of the Executive Order are utter and complete garbage, Just ask conservative Supreme Court Associate Justice Neil Gorsuch, since he wrote the majority opinion in Bostock v Clayton County in June 2022.
Here are the three key contentions in President Trump's "Defending Women" EO: