It's time for a history lesson; followed by a cold, hard truth that many Americans want to shut their eyes to.
First, the history. American Presidents have never bowed to foreign heads of state (and, classically but not as a rule, American citizens in general haven't either). It is not only tradition, but a matter of close-held principle. One finds the theory for this in the Constitution, Article 1, Section 9, Clause 8:
No Title of Nobility shall be granted by the United States.
Now, this may, at surface, seem like a simple matter of our United States not granting its own citizens the titles of Duke, Lord, etc. to avoid the entrapments found under British rule; and to an extent that would be correct. However, the essence of this law reaches much further than the letter of it. In order to understand a law, one must first understand the moral that the law is supposed to implement. The best quote I could think of (though a myriad of more obscure quotes exists) comes out of the Declaration of Independence:
All men are created equal . . . endowed by their Creator with certain unalienable rights.
The two major facets of this quote are the beliefs that: 1. All people are created beings, and 2. The Creator made all people fundamentally equal; i.e., having the same, basic rights. This belief is often expressed today in the concept of "human rights." The fact that most believers in "human rights" seem to eschew the advent of a Creator, though inconsistent in my opinion, does not change the fact that human beings are seen as equal amoungst themselves.
Flowing from this idea (or, as I would put it, truth) that human beings have equal rights, a title of nobility cannot be imparted as it would grant the owner of said title special, additional rights. It would, in essence, claim that noble as a superior human to his or her fellows. From this idea, the founders wrote Article 1, Section 9, Clause 8.
Logically, one cannot deny a title of nobility to one's countrymen on the basis that all men are created equal, and then turn around and recognise a title of nobility for a foreigner. That is why the Presidents of the United States have never bowed to a foreign power. Not only would it signal our entire nation's fealty to that foreign leader (though not through policy; only symbolically), but it also recognises that leader's superiority to the common man. This is about as un-American as it gets, barring a reversion to slavery.
Flowing from this are other ideals and actions, such as the US flag dipping to no nation. Many globalists (and ignorant Americans who wish to sound "cultured") simply view this as American hubris, but this misses the mark by a long shot. The refusal of our armed services and Olympians to dip the United States flag to another country, leader, etc. stems from the idea that the United States stands for all free men, and that free men do not bow to anyone but God (and, even then, at their choosing). It symbolises the reality that no men are born better than their fellows, and that certain rights are innate to all men; handed down from their Creator.
So, that's the basic history. Now for the reality.
Many (if not most) people in America take the exercise of their rights as a given. They believe that the ability to speak, worship, and express themselves freely is a natural thing, and that it is a permanent advent. They often, as a result, believe the poeples of other countries to be inferior, weaker, or "differently righted" than those of this country. Each of these conclusions are antithetical to the ideal we've discussed thus far. These conclusions are what enable people to say things like, "Oh, the Chinese just live in a different culture. Freedom wouldn't work over there." The illogicality of this statement is so manifest that I won't here enumerate it; suffice to say, all people would wish to be delivered from the barrel of a gun if they're facing it.
Note that above I said "exercise of their rights," not simply "rights." Remember, rights are innate and handed down from our Creator. They are not dictated to man by man, but discovered by man in man. The exercise of these rights is proper and correct for mankind. From this, we logically can surmise that all people, regardless of geography or the flag they live under, are entitled to these rights (life, liberty, pursuit of happiness).
Why, then, are the people in China, or Vietnam, or North Korea, or Cuba, or a slue of other nations not able to exercise these rights? Why can we in America? The conclusion that Americans are innately different than their fellow humans around the globe is unsatisfactory. If that were true (that we as Americans "just won't be pushed around"), then there does exist a class or caste hierarchy amoung men, and rights are not equal across the board. We reject that premise.
To what, then, do we attribute this global disparity? The simple answer is usually the best answer. In this case: government. There exists a uniform set of rights amoung people; there is not a uniform government. Now, I'm not bucking for global government here. I think each nation ought to rule itself, and that local government is the best government. But, I would like to throw out the idea that men live under different rules because of the different governments they live under. Of the democracies and republics, the rights to free expression, religion, and speech are generally exercised with little impediment. Of the communisms, despotisms, monarchies, etc., those rights are usually limited to a greater or lesser degree. And why not? If nobility exists, in whatever form, why not unequal rights? It is logical, but it argues from a faulty premise.
Here's the truth Americans generally don't want to look at: the exercising of our rights is not guaranteed. Neither is it a permanent part of the landscape. If we are, indeed, men like any other; and if, as we know is the case, men elsewhere are denied the free exercise of these rights: therefore, we can be denied the free exercise of these rights. If other men, whose freedoms are repressed, live under repression because of their government; therefore we could live under repression with the same form of government as them.
We as Americans need to realise that our rights are granted; not guaranteed. Like a garden, our government must be tended, weeded, cared for, and protected to flourish. Our purpose in managing our government, just like managing a garden, must be deliberate and with a clear goal in mind. Are we tending the garden, or have we let the weeds from within and the elements from without hinder its purpose?