There is a great deal in this question. The first point I would make is in the wording of the question. If bound to the condition of relativism, no one would have the right to make any moral judgments about another culture. Relativism forbids anyone from doing so. Relativism makes the presumption that moral judgments lead to action. These moral judgments, it is believed, lead to a denigration of another culture. (A case can be made that this presumption is, in itself, a judgment.) Yet, if a culture is bound to upholding the tenets of relativism, then it cannot morally judge another culture.
It is from this point where the question leads to murkiness. Cultural relativism, no matter how it is enforced, cannot prevent nations and societies from feeling particular ways. If it did this, it would be guilty of the same absolutism that it criticizes. Cultural relativism cannot force individuals into silence. Hence, even if a nation like America found itself wedded to the idea of relativism, this position does not prevent members of its society to make judgments. Cultural relativism does not stop individual thought or belief as it is meant to encourage it. Hence, within the thoughts of American culture, it does have a right to make a judgment. From a philosophical point of view, American culture might be able to do little with it, but it can make judgments against another culture. This is done all the time. America makes statements when it sees something happening that it deems as wrong. When there is bloodshed in Syria, for example, American society, through its government, voices its opinion. It is practically impossible to demand that American culture not speak. It might be bound by strands of relativism to do anything about such conditions, but relativism cannot demand that a culture not speak. In doing so, an absolutist position is reached which is the very opposite of the philosophical strand.
Finally, I would suggest that there is a good argument to be made that American History is not bound to cultural or moral relativism. While there is much in our history that preaches tolerance and relativist brands of acceptance, our Constitutional background argues from an absolutist point of view. The Constitution demands that there are specific goals that must guide the nation. The nation is free and compelled to speak to these ends. Forming a more perfect union, securing the blessings of liberty to ourselves and our posterity, and establishing justice are a few of these. Such bedrock goals are not relativist. They are quite forceful and absolutist in their scope. These ideas are what helps to guide American voice. When the nation witnesses these ideas not being embraced in different parts of the world or even domestically, it speaks. The historical DNA of our nation suggests that we do, given our background, have the right to make moral judgments. When rights abuses are committed, we have the ability to speak because our own nation stresses that these are not permitted. When women are being beheaded by the Taliban, the American sensibility is not to retreat to relativism. We speak out against this. We say something is right or wrong. In this light, our historical background and founding help to give a voice that can morally judge another culture. Yet, it is in this point in terms of what can be done with this voice where obscurity seems to be present.