No, it's not. Chief Justice Waite did not cite de Vattel at all. His writings appear nowhere in the Court's unanimous opinion.
The "nomenclature" referenced would therefore be something other than English, since Waite's description of the particulars of natural born citizenship do not correspond to English common law.
Which would be, what, exactly?
I could attempt to paraphrase this several more ways, if that would help. I'm accustomed to doing so when dealing with offshore suppliers in China and Korea. They find it helpful. Do you?
Good to know that you're a helpful fellow with your Asian trading partners, but paraphrasing won't be necessary here. Pointing out what "something other than English" is, however, would be great.
Looking at the Constitution itself we find that it was ordained and established by 'the people of the United States,'3 and then going further back, we find that these were the people of the several States that had before dissolved the political bands which connected them with Great Britain, and assumed a separate and equal station among the powers of the earth,4 and that had by Articles of Confederation and Perpetual Union, in which they took the name of 'the United States of America,' entered into a firm league of [88 U.S. 162, 167] friendship with each other for their common defence, the security of their liberties and their mutual and general welfare, binding themselves to assist each other against all force offered to or attack made upon them, or any of them, on account of religion, sovereignty, trade, or any other pretence whatever.
Whoever, then, was one of the people of either of these States when the Constitution of the United States was adopted, became ipso facto a citizen-a member of the nation created by its adoption. He was one of the persons associating together to form the nation, and was, consequently, one of its original citizens. As to this there has never been a doubt. Disputes have arisen as to whether or not certain persons or certain classes of persons were part of the people at the time, but never as to their citizenship if they were.
Additions might always be made to the citizenship of the United States in two ways: first, by birth, and second, by naturalization. This is apparent from the Constitution itself, for it provides6 that 'no person except a natural-born citizen, or a citizen of the United States at the time of the adoption of the Constitution, shall be eligible to the office of President,'7 and that Congress shall have power 'to establish a uniform rule of naturalization.' Thus new citizens may be born or they may be created by naturalization.
The Constitution does not, in words, say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their [88 U.S. 162, 168] parents. As to this class there have been doubts, but never as to the first. For the purposes of this case it is not necessary to solve these doubts. It is sufficient for everything we have now to consider that all children born of citizen parents within the jurisdiction are themselves citizens.
There is ample reference to practically all the above in The Law of Nations. It's clear to me, that during the course of the Revolution against the English and the intervening years under the Articles of Confederation, that key elements of The Law of Nations were adopted and internalized, coming to be regarded as the "nomenclature" to which Justice Waite referred. He clearly was not referring to any English common law "nomenclature" because the terms and qualifiers do not coincide nor do they agree with what he wrote.