Because Trump seems to be treating Executive Orders like they are legal decrees of a monarch, it is worth reviewing what Executive Orders are.
In the U.S. what counts as "law" is really broad. It encompasses a lot of things, statutes passed by Congress, judicial decisions, regulations from administrative agencies, etc. Executive Orders are treated as a kind of law. They can be legally binding. But they are not like statutes or regulations in that they aren't really binding on anyone who does not work in the executive branch of the federal government. At least not directly. An Executive Order is just an instruction to the executive branch from the President, the head of that branch, how to carry out some executive branch function. So, for example, the President can instruct the executive branch of the federal government to start referring to the Gulf of Mexico as the "Gulf of America" in its communications and publications. So when the U.S. Geological Survey comes out with its next map it will say "Gulf of America" and documents about the Louisiana Coast will start using that term. But that doesn't mean that anyone anyone else has to use that stupid name. It certainly doesn't require Google to change its maps. If Google does make that change, that is on Google.
Also Executive Orders, while treated as legally binding, they are the least binding of any form of law in the federal government. The priority for laws in the federal system is like this: Constitutional Provisions > Statutes > Regulations > Executive Orders. So if an Executive Order says something that conflicts with any other kind of federal law, that other Federal Law would prevail. For example, an Executive Order may say that the federal government should interpret the words "sex" or "gender" to mean the sex assigned at birth, and that would be binding on the federal government... except if there is a regulation from a prior administration that has not been rescinded yet that defines those words differently. (Except, of course, in the Judicial Circuits with Judges who blocked those rules)
Even though Executive Orders only directly apply to people in the executive branch, they can definitely have real-world consequences for people who do not work for the government. How members of the government interpret and apply the law can have widespread consequences for sure. But keep in mind that Trump's Executive Order pen is not as all powerful as he pretends it is.
