Actually it isn't settled law, as it has never been addressed by the U.S. Supreme Court. The closest the court has gotten was United States v. Wong Kim Ark, 169 U.S. 649 (1898), which many including myself see as distinguishable in that the U.S. Supreme Court held that the parents of a child need to have a permanent domicile and residence in the United States (which I do not believe people illegally in the U.S. or there only on a student or tourist visa have.
Further even earlier in the Slaughter-House Cases, 83 U.S. (16 Wall.) 36 (1873), (a consolidation of three similar cases) the U.S. Supreme Court, stated, albeit in dicta, that "the phrase 'subject to its jurisdiction' was intended to exclude from its operation children of ministers, consuls, and citizens or subjects of foreign States born within the United States"
Perhaps this is an attempt to get the issue before the U.S. Supreme Court.
Even if this were a truly legitimate way to address the issue, an Executive order could be undone by a subsequent President.
All of this said the United States and Canada are the only two Western states where a child born to parents who are not themselves legally in the county or who indeed do not have a right of permanent residency is automatically a citizen.
So in the greater scheme of things the President is correct.