It may be the day the Declaration was signed, but it certainly didn´t make the "United States" a country yet. There was the whole mess of five years of war against the British first. Even after Cornwallis surrendered, it took over six years for the Constitution to be ratified, and almost another year after that before it went into effect on March 4th, 1789. There was a lot of bickering and infighting as to the makeup of the country-to-be, as the small states like Delaware were apprehensive of being drowned by the bigger states like Virginia (hence two Senators for each state, regardless of size or population--no one in 1787 envisioned a North Dakota or a Wyoming). July 4 is a convenient date for a holiday, but it certainly didn´t represent the day the USA came to exist as an acknowledged sovereign country.
But early March is not ideal for picnics or fireworks in much of North America, so July 4th it is. Once again, my wife and I will probably go down to Köln and celebrate with our German friends of the Amerika Haus, which is a bi-national cultural foundation started after World War II. Thomas Jefferson remains my philosophical hero of the 18th century, and to have written the Declaration of Independence at age 33 was no small feat.