On September 17, 1787, the United States of America was born with the signing of the Constitution. The Declaration of Independence is the date of conception; the Articles of Confederation mark the first detected heartbeat; and ratification in 1789 is when the first breath was drawn (or maybe shortly thereafter when Congress was elected and first met).†
OK, so it's an overextended metaphor, but illustrates well that we have a living Constitution. That you are reading this via the internet (even if someone else printed it for you and you never saw it on the actual blawg) demonstrates that pretty well; there are lots of failures of imagination in the document, particularly the narrow, literally-limited (or original-public-meaning limited) list of what is protected from government intrusion that didn't even contemplate electronic communication. These words, for example, have never become a "paper" while in my possession — not as you receive them, not in draft here. That's just a particularly easy example.
tl;dr Context matters, including our choices of what to call a "birthday." That we often celebrate when we yelled abuse at the sovereign instead of when we actually gave birth to something new bears some consideration, too.
† We'll just slide right by the later gender-fluidity-like incidents like the Reconstruction Amendments and Nineteenth Amendment. Bringing in the gender-reveal parties apparent in Shay's Rebellion would be going more than a bit too far. Let's just celebrate the real birthday, ok? Perhaps by using the fingers we have left after the uncontrolled fireworks at the FAKE birthday in July to vote?