No, it would not.
The Birther argument is two-fold.
Part one is that he was not born in the US.
Part two is that he is not born of two US citizens (under their completely made-up theory of what a natural born citizen is). Producing a birth certificate would do nothing to end discussion on part two of their theory. And every major birther seems to subscribe to this theory.
So, if the birth certificate says he was born in Hawaii to a Kenyan and an American, do the birthers go away? Of course not.
Actually there is a third rail: the massive file of hidden from the public documents referencing his life as an Indonesia, college entry records, grades, tuition paid by whom (we know some of that came from Islamic sources), and the record of his leagl names.
Completely made up?
What about this 1758 treatise on natural law that was the most cited such work, more so than Blackstone's commentaries, in the first decades of the Republic? (The original was in French, but many of the founders, Jefferson, Washington, Jay, and of course Ben Franklin, could read French, so they did not need the English translations.
Les naturels, ou indigenes, sont ceux qui son nes dans le pays, de parents citoyens(diacritical marks left off)
Or in English, with some more of the translation for context. (It was painful typing in the French, but you can view it here)
§ 212. Of the citizens and natives.
The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives, or natural-born citizens, are those born in the country, of parents who are citizens. As the society cannot exist and perpetuate itself otherwise than by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights. The society is supposed to desire this, in consequence of what it owes to its own preservation; and it is presumed, as matter of course, that each citizen, on entering into society, reserves to his children the right of becoming members of it. The country of the fathers is therefore that of the children; and these become true citizens merely by their tacit consent. We shall soon see whether, on their coming to the years of discretion, they may renounce their right, and what they owe to the society in which they were born. I say, that, in order to be of the country, it is necessary that a person be born of a father who is a citizen; for, if he is born there of a foreigner, it will be only the place of his birth, and not his country.
Note than in the French the section 212 title is "Des citoyens et naturels." Which would really translate as "The Citizens and naturals or natural born citizens".
The work is of course " THE LAW OF NATIONS OR PRINCIPLES OF THE LAW OF NATURE APPLIED TO THE CONDUCT AND AFFAIRS OF NATIONS AND SOVEREIGNS FROM THE FRENCH OF MONSIEUR DE VATTEL."
You see "nature" "natural born". The "born in the country is enough" (actually born in the King's lands) criteria for Natural Born Subject, was of course set down by the Kings and Queens, whose interest was to have lots of subjects, very few of which stood a chance in hell of becoming King or Queen.
But I guess some birther invented a time machine, went back to 1750s Germany (Vattel was Swiss but was working in what is now Germany) and convinced him to write that, just to legitimize B.H. Obama. The racist bastid!