The law, 8 USC 1401 and 8 USC 1409, which covers out of wedlock births) doesn't cover Natural Born citizenship, but that isn't even what it says for citizenship at birth. For one US parent there are restrictions as to the residency period of the parent in the US (now 5 years, 2 of which must be after the 14th birthday, but in 1961 it was 10 years, 5 after the 14th birthday, which Obama's mother did not satisfy at the time of his birth. So, if born outside the US, he's an alien, not a citizen at all) For those with two US parents, it only requires "a residence" in the US for one of the parents.
But such persons born outside the US are not even native born, they are naturalized at birth, per Supreme Court rulings, and thus are not eligible to the office of President. They don't even have 14th amendment status, since they were not born in the US, nor naturalized in the US.