Assuming you mean the first person born in the United States at the point it became its own independent country, I'm afraid this question may not be possible to answer with definitive accuracy.
One problem is that it's difficult to define exactly when the United States became a country. Was it on July 2, 1776 when the Continental Congress voted to declare indepedendence? Was it on July 4 when the Continental Congress approved the Declaration of Independence (document)? Was it on September 3rd, 1783 when the Treaty of Paris was signed? Or was it on January 14th, 1784 when the Treaty of Paris was ratified by the United States?
We can pick one of these dates if we'd like, of course. Culturally, most Americans would agree on July 4th, 1776 as the "birthday" of the nation. Even then, though, it would be hard to pinpoint exactly one person as the first to be born in the United States, because I'm not sure we can pinpoint the exact time of day America became independent and find record of a baby born immediately after that time.
Edit: I will add that we do know that Virginia Dare, born August 18, 1587, was the first child born of English decent in North America. I believe this would make her the first person (excluding natives) to be born within the territory that would later become the first part of the United States two centuries later. Even if that doesn't quite answer your question, maybe you'll find it interesting anyway.
Your question could be adjudged in a few different ways.
The first person born in the lands that are now called the United States.
The first person of European decent that was born in the land that is now called the United States.
The first person born after the United States declared independence.
The first person born after the United States was recognised.