The Founders embraced..domestic matters in Vattel's Law of Nations. Domestic matters: chapter XIX: natural born citizens are born to citizen parents. A country cannot perpetuate itself unless its citizens are born from citizens. 
But you are FIBBING BIG TIME. Because if you find a link to the book you are quoting, which YOU conveniently left out, (I wonder why???)then you see that it wasn't Vattel's quote about citizens the No.5 footnote was about. OH NO!!! Here is the link to the book:
And it is what it was talking about:
However, "international law, (,em>jus inter gentes) is only a subset of the law of nations (jus gentium). Indeed, a large segemt of the law of nations was the lex mercatoria that governed private commercial transactions. The Founders would not have viewed the law of nations as a body of law governing only international subjects. In the next chapter, we will examine the positive law of nations.
Hmmm, COMMERCIAL LAW, not citizenship stuff. And here is the quote the No.5 footnote was referring to, and WHAT A SURPRISE, it is not about natural born citizenship at all: This is from Vattel, since the book you quoted chopped it up a little
The state is obliged to defend and preserve all its members (§17); and the prince owes the same assistance to his subjects. If, therefore, the state or the prince refuses or neglects to succour a body of people who are exposed to imminent danger, the latter, being thus abandoned, become perfectly free to provide for their own safety and preservation in whatever manner they find most convenient, without paying the least regard to those who, by abandoning them, have been the first to fail in their duty. The country of Zug, being attacked by the Swiss in 1352, sent for succour to the duke of Austria its sovereign; but that prince, being engaged in discourse concerning his hawks at the time when the deputies appeared before him, would scarcely condescend to hear them. Thus abandoned, the people of Zug entered into the Helvetic confederacy.* The city of Zurich had been in the same situation the year before. Being attacked by a band of rebellious citizens who were supported by the neighbouring nobility and the house of Austria, it made application to the head of the empire: but Charles IV.73 who was then emperor, declared to its deputies that he could not defend it;—upon which, Zurich secured its safety by an alliance with the Swiss.† The same reason has authorised the Swiss in general to separate themselves entirely from the empire, which never protected them in any emergency: they had not owned its authority for a long time before their independence was acknowledged by the emperor and the whole Germanic body, at the treaty of Westphalia.
Vattel's Law of Nations chapter 17
Sooo, why did YOU try to fool everybody here with a phony quote where it wasn't about what you said, and then didn't even leave a link??? Aren't YOU ashamed of yourself???