That is, however, only a surface reason for choosing that date, and as point of fact it is not even the day itself that matters, but rather the ideals commemorated on this day within that essential document, that are at the heart of the matter. While the practical considerations upon which the Declaration itself rests are necessary (the "long train of abuses" section that is so important in understanding the basis for the colonists' argument), they would carry only subjective weight on their own without the objective standard which the Founders recognized as binding upon all people. Allow me to exerpt the particularly relevant statements from the Declaration of Independence itself, which you can read, among other places, at Yale's Avalon Project:
When in the course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the laws of nature and of nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
We hold these truths to be self-evident:
That all men are created equal; that they are endowed by their Creator with certain unalienable rights; that among these are life, liberty, and the pursuit of happiness; that, to secure these rights, governments are instituted among men, deriving their just powers from the consent of the governed; that whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness.
One must note that the ideas about humanity and liberty are somewhat at odds with much of Modernity. On the one hand you have absolutists like Hobbes who view liberty in binary terms--either a person holds it and is completely autonomous, or he cedes it entirely to the State ostensibly for his own protection (though in reading Hobbes one could be reminded of the sort of "protection" offered by a Mafia Don and his henchmen). There exists no right of revolution in such an understanding of liberty; no idea of the State having to be dissolved if and when it becomes destructive to natural liberty.
On the opposite end of the spectrum, you have the notion of liberty as humankind's freeing himself from his attachments to unnatural things like property, which is the sort of idea seen in Marx or certain other (mostly German) Rational Historicists. This is liberty construed as being who you ought to be, not becoming who you are, but in doing so it misconstrues the whole world of experiences within which humans must operate by necessity as temporal creatures. There is a third alternative view of liberty that comes in at the very end of Modernity; that it is merely a construct of human life much as anything else; that liberty is whatever we say it is (or is not), contingent upon the assertion of our own will. This last notion, however, is perhaps too far-flung from the world in which the Founders operated, and so is not really relevant to this topic.
The notion of liberty as seen in the Declaration and other writings of certain Founders, is almost Classical (or at least Neoclassical) by comparison to those mentioned above. When Jefferson and the others on the committee for writing the text of the Declaration wrote that:
Whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness"
They echo the words of Plato in Book IV of The Laws:
But anyway, in our consideration of the nature of the land and the order of the laws, we're looking now to the virtue of the regime. We do not hold, as the many do, that preservation and mere existence are what is most honorable for human beings; what is most honorable is for them to become as excellent as possible and to remain so for as long a time as they may exist.
The preservation of a particular political order is insignificant, then, by comparison to the preservation of the ends of good living, and if the regime is no longer suitable to this then it must cease to exist and a new order established in its place, one that sufficiently provides for these ends.
Of course there are Modern aspects to the American Founding as well, particularly with respect to the idea of the State as something created entirely by humans rather than as part of a natural/historical/divine process. After all, the interlocutors in the above cited Platonic dialogue were on their way up to the temple of Zeus on Crete where the mythic lawgiver had handed down the laws to Minos, as the proper place to offer prayers and sacrifices in setting out on an endeavor to form a new colony.
We do see some homage to that divine language in the Declaration (the four references to the deity being: "Nature's God," "Creator," "Divine Providence," and "Supreme Judge of the World"), though we don't get the same feeling here as we do from the ancients that the hand of the divine is what establishes a new regime, and in point of fact it is the people themselves who are throwing off the chains of the old oppressing order and setting forth a new one...with providential blessings being asked for of course. And yes, it is extremely important to the idea of natural rights that we understand they do not come from any human or institution made by humans, but rather are innate and able to be understood equally through natural reason.
So there is a Modern and a Classical part to the American Founding, which is why it is often said to be the bridge between the two (often disparate) views on politics and liberty. I wouldn't call it perfect by any means, particularly since we've seen the need to modify it as our understanding of the application of liberty has evolved over the many years, and clearly the reasoning set forth in the Declaration holds true that should the regime be irredeemably hostile to the rights of her citizens that it ought then be cast off in favor of one that is not. Nevertheless, it is definitely worth celebrating the ideas and the affirmation of them upon which this regime is based. And with that I wish an excellent Independence Day to everyone.