For that answer, I refer you to the Law of Nations, which is contained in Article 1, Section 8, Clause 10 of the US Constitution.
BOOK I. OF NATIONS CONSIDERED IN THEMSELVES
CHAP. XIX. OF OUR NATIVE COUNTRY, AND SEVERAL THINGS THAT RELATE TO IT.
§ 215. Children of citizens born in a foreign country.
It is asked whether the children born of citizens in a foreign country are citizens? The laws have decided this question in several countries, and their regulations must be followed.(59) By the law of nature alone, children follow the condition of their fathers, and enter into all their rights (§ 212); the place of birth produces no change in this particular, and cannot, of itself, furnish any reason for taking from a child what nature has given him; I say "of itself," for, civil or political laws may, for particular reasons, ordain otherwise. But I suppose that the father has not entirely quitted his country in order to settle elsewhere. If he has fixed his abode in a foreign country, he is become a member of another society, at least as a perpetual inhabitant; and his children will be members of it also.
The citizenship of the father determines the citizenship of the child, so Obama is NOT a natural born citizen according to Constitutional law.
The specific clause says: To define and punish Piracies and Felonies committed on the high Seas, and Offences against the Law of Nations.
Now, the clause (sub-clause, actually, if that) says nothing about the U.S. being bound by the "Law of Nations" -- and, indeed, we are not. The clause (if you really want to call it that....) does nothing more than authorize Congress to "define and punish ... Offences against" it.
Further, it's pretty clear that the "Law of Nations" does not refer to any specific body of internationally promulgated law; rather, it is more a reference to a set of philosophical principles -- which is why Congress was given power to define offenses against it. A good discussion can be found here. In particular:
In the debate on the floor of the Convention, the discussion turned on the question as to whether the terms, ''felonies'' and the ''law of nations,'' were sufficiently precise to be generally understood. The view that these terms were often so vague and indefinite as to require definition eventually prevailed and Congress was authorized to define as well as punish piracies, felonies, and offenses against the law of nations.
It is, further, quite clear that the context of the term as used in the clause you cite is much different from any questions of Obama's citizenship.