I haven't read the text of the law yet, but I would not be overly surprised to see some details in there somewhere that stipulate that the law only applies if you identify in certain ways. We do not have a forward-looking judicial branch here.
Carl, it's a Supreme Court decision not a law. And as they go, fairly short; 27 pages for the decision, the rest of the 100 pages are the dissent. The decision was written by a Trump appointee, Neil Gorsuch, who, in the tradition of his mentor, Antonin Scalia, considers himself a textualist. That is, the court went back to what the words of the law in question say.
The law in question is Title VII of the 1964 Civil Rights Act (I got the title wrong in my earlier post) covers employment discrimination so the standard he applied is "except for sex." So, it doesn't come down to, for instance, 'If women are allowed to wear pants on the job, men are allowed to wear skirts;' the ruling says, 'If women are allowed wear skirts on the job, so are men.'
I throw in that nuance to suggest that if, as a matter of safety or uniform, a company says all employees are to wear pants; then so it shall be. But they can't say only women may wear skirts.
As to identifying oneself as anything other than a human being and an employee, I would be surprised if there was anything of the sort in this ruling because one of the things textualists insist on is not rewriting laws -- that's Congress's job. They also stress simplicity. And in 1964 no one had yet cooked up our current alphabet soup of gender identities. So, even though I haven't read the ruling, I'd say you probably DON'T need to label yourself (I'm sorry I hadn't thought my statement through in my earlier post through.) to wear a skirt or dress to work.
Strictly as a matter of practicalities I would suggest two things: 1) give your supervisor a heads-up before you take the plunge, because they're categorically allergic to surprises and 2) if your employer has a diversity policy, have one of its gender identity labels that you're comfortable with handy, just in case you get questioned about it being within company policy. If your company policy has you covered, why get into a debate about current events? These days most supervisors aren't terribly concerned about what their employees wear to work, just so long as it's appropriate to the environment, but none of them like pains in the neck.