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Were I teaching high school, I think you might be one of my dream students. This is an outstanding question. It's also massive since you're really asking how Americans have justified their atrocity-heavy history and congratulated themselves on being the land of freedom, rights, and progress at the same time. Every nation probably does a poor job of living up to their PR over a long enough timescale, but Americans tend to imagine themselves as different and united by a common creed that justifies taking themselves as the permanent exception. Of necessity, I can only answer for the atrocity I've studied. So let's talk about how Americans square freedom with slavery.

Initially, they felt no need to do so. When we speak of "rights" today, we usually mean something like constitutional rights or human rights, which a man told us are inalienable whilst pursuing for his entire life the alienation of them for his personal profit. Likewise the other men who spoke passionately in favor of freedom, who fought for it in the Continental Congress or the Revolutionary army, have no shortage of slaveholders among them. The formula may be that they fought for rights which belonged to everyone, endowed by their Creator and all that, but their actions show something else. Rather they fought for a narrower kind of rights, more like the rights to vote (only for men of considerable property, which as a practical matter ensured almost only for white men too) or the right to make a film adaptation of a novel.

Movie rights and voting rights, because of how political thinking has evolved for the past few centuries, make for an odd couple. The whole point of buying the rights to produce films of Harry Potter, Lord of the Rings, or this or that superhero is that the owner gets to do that and no one else does. Such rights are fundamentally exclusive and private, or they are not rights. Having the right is the power to do, but also at the same time the power to prevent anyone else from doing the same. The vote, or more broadly a right to have a say in governance, we now imagine as universal at least among citizens of age. The framers didn't live in that world and most of them did not much like the idea.

This is all a long way to tell you that while the American Revolution's famous documents and rhetoric promise a universal freedom, it is universal for people just like them. Maybe other white men (but only white men!) can get in on it over time, which is something that happens over the course of the first half of the nineteenth century, but that's where white America draws the line. As a matter of fact, it's precisely where the line gets drawn.

Because this is a process, we can pick almost any starting point. There's a good argument that we need to begin talking about white American justifications for slavery -black Americans are, for obvious reasons, not prominent on the justifying slavery bandwagon- about the same time as we start talking about slavery in America in itself. That could push us back to the 1620s. There is discussion and discontent with slavery on a low level, usually on religious grounds, almost that far back but it's a relatively minor element of white though confined to lonely, isolated voices. Over the eighteenth century, that changes. There's no single reason why, but the idea of rights gets more universal and people of the framers' era understand the contradiction. Hence the famous hand-wringing that Jefferson does and the post-revolutionary wave of manumissions (enslavers giving the people they enslave their freedom) and emancipations (legal termination of slavery, usually done over a protracted period and only for people born after a certain date).

Because some white Americans in areas where slavery is marginal, and a few who agree with them in areas where it is not, decide that slavery is bad and needs to go, other white Americans need arguments to defend it. These evolve over time and in tandem with conceptions of racial difference, all interweaving together in complicated ways. It's conventional to simplify it down to the Necessary Evil argument and the Positive Good argument, maybe with an inclusion of the growth of scientific racism in the middle nineteenth century. Then one further breaks it down with Necessary Evil coming first and Positive Good taking over in a reaction that plays out over the 1820s and 1830s. The record actually shows positive good and necessary evil themes coexisting from the start and whether or not one argument eclipses the other, and when, tends to be a question of who we're talking about and where. But that's all hugely complex and we can treat the standard version as a decent first approximation, to which I'm going to bolt on the necessary racial thought because the story isn't complete without it and it's an important place that my answer about slavery and /u/snapshot52 's excellent one about the genocide of the American Indians intersect.

White Supremacy When we talk about "racism" in US history, we're almost always stalking about white supremacy. At its base, white supremacy is the conviction that some people who deem themselves white believe themselves better, more deserving, and generally entitled to commit tremendous injustices upon the lives of people they deem non-white. They either then commit those injustices or, more often, have already been committing them but now feel more insulated from criticism by people who matter to them. This is almost never the actual victims except insofar as it involves ensuring that the victims don't have the means to suddenly insist that they matter, whether through violence or access to the political process.

White Americans in the founding generation do not, even among those who become uncomfortable with or even the smaller number who work for its end, think of black Americans as their equals. They are inferior on multiple grounds which might be rhetorically temporary but are de facto permanent. Consider, for example, the first naturalization act the United States ever passed. Qualifications for US citizenship, effective March 26, 1790, are as follows:

any alien, being a free white person, who shall have resided within the limits and under the jurisdiction of the United States for the term of two years, may be admitted to become a citizen thereof

Literally from the beginning, this is a white country so far as white Americans are concerned. Slaves cannot be citizens on account of lacking freedom, but a black American who lived in the US for their entire life is excluded. That's the performing of white supremacy, but all the talk about equal rights and freedom has convinced enough white Americans that they need a justification that by this point they have made some largely to defend slavery from white criticism.

Eighteenth century Americans are not scientific racists in the vein of nineteenth century Americans, but they're very much white supremacists. Earlier, religious justifications for slavery (which come down to Africans not being Christians) are by no means gone but have an obvious expiration date and are undermined by well-established laws that conversion to Christianity does not bring freedom in the material realm. They make note of superficial anatomical differences, but don't go in for brain sizes as their children and grandchildren will. They believe that there are racial differences, but those come down to environment.

Environmental racism holds that the tropical African environment has done something to black people which gave them their distinct appearance. They presumably looked white, or at least some intermediate shade, before that. The African sun beat down on them and also caused behavioral predilections which explained why they didn't act like everyone agreed they should: as diligent, faithful, modest, white Europeans. This made them inferior, but also meant that in principle they could change in a new environment. Antislavery whites glommed on to one or two black men who had a skin condition that gave them light-colored patches as evidence that it was happening. Some went farther and argued that black skin itself was just a variant form of leprosy and so might be cured in a single lifetime with the right medicines.

But for the most part, environmental racists don't think that's going to happen. At the very least, black Americans will need generations and perhaps centuries to change. That's functionally, if not rhetorically, the same as saying they're like this forever. Since they are, then it's a-ok to treat them as lesser beings.

Come the nineteenth century, the discussion shifts to much more fixed and internal traits. Skull size (bigger is better) and shape (taller is better) take on great salience. You have naturalists going around getting skulls and filling them with lead shot, buckwheat, and other stuff that they then pour out into something to measure its volume. They then use this volume, which turns out to be smaller, as evidence that black people have smaller brains and so just can't reason on the same level as whites. You could give them the vote, if you were out of your mind, but they wouldn't know what to do with it. They are, by their biological nature, permanent inferiors and so in accord with nature they must have an inferior role in civilization.

Oh man, do I have just the information for you! I want to commend you for wanting to do some digging on this and being willing to push the boundaries with regards to what you might've been told.

Probably the shortest way to answer your question is that the justification varied. But first, let's establish this (because I usually get people wanting to oppose this point): genocide(s) was committed by the United States government against American Indians. With that, let's look at some of the rationale. In this answer, I approached a similar question from several perspectives to highlight that there isn't a single act of justification, but that it changed overtime to accommodate the expansion of the United States.

Terra Nullius

Terra Nullius was the beginning of a doctrine rising from edicts by the Pope during the Age of Exploration. The phrase roughly means "empty land" and was established to justify the expansion of Christian nations in Europe. Along with several other edicts, known as Papal Bulls, these nations declared any lands they "discovered" as under their rule, regardless if people lived there or not. This is the formulation of the Doctrine of Discovery (DoD). This doctrine also stipulated that if people were found on discovered lands, they did not have any title to said lands, for they were non-Christians. This means that Indigenous peoples, such as Native Americans, were viewed as pagans and not deserving of the same rights and liberties as Christians.

DoD and Marshall Trilogy

Over time, as North America (focusing on this region) was further colonized, European nations, particularly England, grew to view Indigenous group as equal sovereigns, entering into treaties with them and recognizing their title to their lands beyond the boundary of the Thirteen Colonies and other claimed lands in Canada. Naturally, this view did transfer to the United States, which was formalized in 1823 during the Marshall Trilogy of Supreme Court cases. These cases, the first occurring in 1823 under the title Johnson v. McIntosh, recognized a transfer of land titles under the Doctrine of Discovery from England to the United States after the Revolutionary War. The next two cases, in 1831 and 1832, respectively, both acknowledged the inherent political status of Tribes, but sought to undercut that by making them "domestic dependents," effectively bringing them under federal oversight in the eyes of the United States. However, this didn't mean the U.S. commanded total control over there, especially for the Tribes further out West.

Manifest Destiny and Religion

During the middle of the 1800s, the United States was swept with a wave of expansion. This was known as Manifest Destiny, which was the idea that it was the divine right by God that the Americans expand the borders to the Pacific Ocean and claim as much of the land as possible, for it was their "destiny." This ideology formed both religious and political roots, which led to mixed opinions about how to handle American Indians. Several methods would be tried. These ranged from removal and assimilation to all out physical extermination and political termination. To curb the "Indian Problem," as it came to be known, Americans came up with many justifications for these actions.

• The Indians were savages and poor, thus they needed to be tamed and brought into white society so they could assimilate and become proper people and receive help.
• The Indians were slow, barely human, and couldn't govern themselves or the land in a civilized and proper way.
• The Indians were pagans and needed to be Christianized in order for them to be saved.
• The Indians were standing in the way of American expansion and needed to be removed or eradicated so they didn't pose a threat.

Racism

While the concept of "race" is a more modern notion, the belief that different groups of people could be "better" or "inferior" was definitely a factor since the formation of the United States. The very pejorative that Indians were savages because how they looked, not just acted, helps to demonstrate this. Listed in this post are some examples. They are also quoted below.

Samuel George Morton wrote in his 1839 work Crania Americana:

In their mental character the Americans are averse to cultivation, and slow in acquiring knowledge; restless, revengeful, and fond of war, and wholly destitute of maritime adventure (Morton, 1839, p. 6)

American historian Francis Parkman noted in 1898:

"But the Indian is hewn out of a rock. . .Races of inferior energy have possessed a power of expansion and assimilation to which he is a stranger; and it is this fixed and rigid quality which has proved his ruin. He will not learn the arts of civilization, and he and his forest must perish together" (Black & Wiedman, 1976).

Mary Austin writes in Character & Personality:

In taking stock of the mental processes of American Indians one has always to keep in mind the salience of primitive processes, in the emergence, above the rationalizing tendencies, of what is generally known as subconscious mentality. There can be no manner of doubt that in the normal life of primitives there is a preponderance of the sort of experiences which are described as hunches, intuitions, premonitions, clairvoyance, prognosis through dreams, visions and symbolism, and that these experiences lumped together as "Medicine" are generally more trusted by true primitives than any of the processes called intellectual . . . Among the least successfully organized tribes the possession of such "medicine" traits are the true basis of leadership: direction of war and hunting expeditions go to the one whose "medicine has been good," or whose dreams are most significant (Austin, 1933, p. 234).

Contemporary Justifications

Not only were there past justifications, but we have more current ones as well, which are primarily used to excuse the atrocities rather than justify them, though that happens occasionally too. Here and here are two posts I wrote for our Monday Methods feature in where I describe the state of denial that exists around this topic. Some of the reasons are as follows:

• Disease - Ignoring the active factors of colonialism, such as displacement, killing, and removal, disease is used as a scapegoat to say that most Natives died before Europeans even got to them, leading a "bloodless conquest" myth, suggesting Euro-American settlers had little to do with the death of Native Americans.

• Empty Space - As discussed above, Natives were considered to be uncivilized, backwards, and offering very little. Combined with Western ideas of resource maximization (in the form of industrialization) and land use, Indigenous groups were seen as not using the very land they had, thus they were not deserving of keeping it (which is definitely not true). And Christians found it, so they lost it because...reasons.

• Removal - This one is often used as a disguise, a less impact rather than killing. It is proposed the Natives were simply removed rather than outright killed and when they died, it was because of circumstance, not a deliberate act. However, removal was truly the guise to mask the motivation - the U.S. government, and much of the American populace, didn't want Native around and if they died, so be it. Many of the reservation Tribes would be confined too were not ideal places for either their traditional lifestyles or the agricultural desires of Euro-Americans.

• "They lost the war!" - This justification is an oversimplification attempting to explain that Tribes suffered because of war and conquest, the name of the game in their day. However, this is far from the truth. Many of the atrocities committed against Tribes actually occurred outside of a state-of-war context, such as the Wounded Knee and Sand Creek Massacres. The exiling of the Nez Perce to Oklahoma saw the death of many of my people who were not allowed to return to their homes, the Termination of the Menominee Tribe's political status in the 1950s saw the infant mortality rate rise, the 1970s saw the forced sterilization of Native American women which decrease American Indian populations significantly, the war waged against the buffalo to corral Indians onto the reservations established by treaty that guaranteed their safety... All of these, and much more, happened outside of war. There was no conquest, there was no major defeat. All there was were broken promises, back stabs, and capitalization on poor conditions.

Conclusion

What can we conclude? The justifications and excuses varied over time. They were created to suit the needs of the U.S. government and affiliated non-Native interests at their time. The reality is that the ideals set up in the U.S. Constitution, the ones mentioned in the Declaration of Independence, and all the other moral basis for the American lifestyle were all established from the perspective of white, land-owning, European-descent individuals. They wrote those things with those items in mind and they did not, as is clear by their conduct and other writings, intend for it to be extended to all people at the time, especially for American Indians. Thus, they justified it by excluding us from their world and their ideals.

Hey OP, like the previous commenters, I just want to say it is excellent that you aren’t just accepting what is essentially being force fed. Out of curiosity, which part of the US do you reside in? I ask because I have a firm belief that depending on your location in the states, different aspects of our history are given different levels of focus or importance. In California, I can’t recall exactly how much time was devoted because it’s been quite a few years since high school, I do recall a significant amount of time being spent on our atrocities against natives and slaves. (Trail of tears, 3/5 compromise, etc)