I'd want to see a lot more documentation before I'd accept that.
"The law of nations" -- today's "international law" -- isn't a fixed code of laws, and in a matter like this, it probably doesn't prevail over the laws that nations make for themselves.
You can certainly argue that the law of nations -- or Vattel's ideas about it -- prevail over US law and the common law, but judges probably won't agree.
Most of the founders were familiar with the common law, and Blackstone, an important commentator of the common law doesn't have the same definition of "natural born" that Vattel does.
There was, however, the law of nations (as evidenced by the Constitution itself which mentions it), but more important, there are lots of writings of the framers themselves on the subject. One must look to the purpose of the natural born clause for it to make sense. The framers did not want any foreign influence in the president. Really, anyone who engages in such wishful thinking as to imagine that when it comes to the natural born requirement for president, the framers only cared that a baby was plopped out on US soil disregarding foreign parent(s), is uninformed and not being logical.
You have to look to the law of nations at the time of the Constitution. Vattel isn’t the only one who wrote a treatise on it, he’s right. Justice Story did for American International Law and that followed Vattel, not Locke.
What do you mean you’d have to see today’s law on it? Didn’t you read the British Nationality Act of 1948 and see also that Obama already admitted being governed by the laws of Great Britain at the time of his birth?
Doesn’t matter anyway. A natural born citizen couldn’t have included Obama since the law at the time was jus sanguinis, not jus soli. If all you had to do was be born here to run for potus, then explain why children weren’t considered citizens until their parents were naturalized? That happened right after the Constitution was signed and while the framers were still alive. If that’s not what they meant, that would have never been the case. Even the Constitution calls for Congress to make a uniform rule for aliens, which included their families. There was no such thing as dual-allegiances aka dual-citizenship. The framers ruled against that talk upfront.