There is no simple or 100% correct t answer because valid arguments can be made over several dates. Obviously, 7.4.1776, which is the date we celebrate; but what we actually celebrate is the date we declared our independence. That in itself did not render us independent of England. We still had to fight and win our independence. We weren't even formally known as the United States of America yet. That wouldn't come until the ratification of the Articles of Confederation on 3.1.1781. So that could be considered the birth of our country. However, technically, we had still not won our independence from England. That didn't happen until 9.3.1783. So that too could be considered the birth of our country. Personally, to me, the last one makes the most sense. I see declaring our independence like conceiving a baby. And like the baby is not born until it's free from it's mother's womb, we weren't a country until we were free from England.
As far as Palestine, they too declared their independence. In 1988. According to the logic that merely declaring one's independence establishes a country, then Palestine would have been a country since 1988; yet no one considers that to be the case. They formed a government in 1994, yet still not considered a country. I'm not sure if they're recognized now as a country, though they were admitted to the U.N. a few years ago (not sure to what capacity?).