Doodle just read the actual order. While it is written with lots of nice language about how inclusive NYC has been historically, and how pro-active the city has been in protecting the rights of the LGBTQ community, the fact is that this order has effectively made all City-owned/operated bathrooms unisex.
This is so because the order expressly prohibits anyone from questioning anyone's reasons or motives for using whatever bathroom they choose. Here's the exact language: "All employees of City agencies ("Agencies") and all members of the public using City services must be allowed to use the single-sex facility within facilities owned or operated by the City that most closely aligns with their gender identity or expression without being required to show identification, medical documentation, or any other form of proof or verification of gender."
There is no ambiguity in this whatsoever. Thus, by decree, the mayor has declared that every person in New York may now legally use any City-owned/operated bathroom, regardless of what it says on the door, with absolute impunity. Furthermore, anyone questioning them in any way, shape, or form, including anyone within those bathrooms who may object to their presence, risks prosecution under the New York City Human Rights Law, which prohibits discrimination based on gender identity/expression.
EDITED TO PROVIDE THE LINK: http://www1.nyc.gov/assets/home/downloads/pdf/executive-orders/2016/eo_16.pdf
Yup. Use whatever bathroom you want. If someone else in there doesn't like it, that person is the problem, not you.
Everybody can just do whatever they want to do, social norms and mores be damned.