Fourteenth Amendment and Citizenship is a murky Constitutional question...

"person born in the United States, and subject to the jurisdiction thereof"

Let's just say that there have been numerous arguments in the past regarding this particular amendment, and particularly what the phrase "subject to the jurisdiction thereof" actually means.

United States v. Wong Kim Ark suggested than any child born of resident aliens is a citizen. It stopped short, however, of providing blanket citizen status for anyone born in the United States. There have been several instances where the Courts have ruled in favor of children born of illegal aliens being provided many of the rights and privileges of citizenship (such as school funding, census, etc). Some have argued that these court decisions have settled the issue.

Ultimately, however, Congress did write naturalization laws that reaffirmed the language of the fourteen amendment, without doing much to clarify. So there can be some disagreement as to whether court decisions are interpreting the law, the constitutional amendment, or a combination of both.There is therefor much debate as to whether or not Congress would be able to constitutionally decide to legislate a more specific interpretation of their naturalization laws.

To be clear, I do not believe that the President has authority to use an executive action to clarify and amend a law, and most certainly the President does not have the authority to write an executive action that would clarify a Constitutional amendment. He probably has the authority to write a executive action that provides a guidance as to how to enforce these laws, but that guidance would need to be within the confines of existing law, and couldn't (IMHO) simply do a full one eighty on how we are currently enforcing these laws. The exception would be, of course, if that executive guidance were cancelling out another executive guidance that was felt to be outside of the confines of existing law.

I believe that if the President attempted to do so, that he would lose that court case all the way up to the USSC, based on the simple concept of separation of powers and the limits to Presidential authority.

Meanwhile... I certainly have no doubt that new naturalization laws (written by Congress) that undercut the idea of blanket birthright citizen would bring immediate legal challenges, and I have absolutely zero doubt that there would be lower court decisions that would rule those laws are in violation of the 14th Amendment. But the only question here would be how the new USSC would view such an argument.

To the degree that the Constitution is fairly straight forward, I would pretty much always believe that a conservative court would demand that all laws follow that portion of the Constitution. If you want to ban free speech (first amendment) or you want to ban all firearms (second amendment) then you better amend those amendments through a constitutional procedure. If you attempt to write laws that are in clear violation, they will be struck down as such.

But the question lies in where the constitution (or an amendment) is somewhat vague and open to interpretation. Would conservative Justices provide more leeway for Congress to make a declaration as to determine that interpretation, or would those same Justices believe that it's the sole place of the courts to resolve such questions?  Would it be conservative to "defer" to Congress and not take it upon themselves to "legislate from the bench"  or would it be conservative to take constitutional issues out of the jurisdiction of the legislation and make that determination at the Judicial level.

Personally, when the constitution is not "clear" about something I would offer that a piece of legislation that appears to bear a reasonable interpretation of the meaning in question should be considered constitutional. In those cases, I would think that we would like the Courts to defer to the legislative branch rather than supersede Congress on a matter of opinion.

In other words, I believe that the Court's job isn't to prevent our Congress from interpretation of constitutional principles, but rather to prevent Congress from blatantly "changing" the meaning of constitutional principles.

To the degree that some people believe that the Constitution should be "living" or "changing" with the times, those interpretations should "always" be left up to our legislative branch. Large legal and social changes in society should be determined by the branch that is actually elected by those people who make up society, rather than by Judges who are not accountable. To the degree that the courts have a say, would be to reign in changes that run counter to Constitutional issues or legal precedents. Never, should it be on the courts to be the ones that "move" society in a new direction, especially if that direction competes and counters with Congressional laws written by elected legislators, or with other sorts of referendums and state amendments.

So it's unclear exactly what the Roberts court would do with this if Congress wrote a law clarifying birth right citizenship. My gut tells me that they would look strongly at the legal precedent (that favors the idea of birth right citizenship) and argue that the 14th Amendment provides birth right citizenship, but I wouldn't be shocked (or upset) if they choose to let Congress settle the question.