Native Americans were enslaved in the Americas, though you could be forgiven for not knowing this, as it is a part of the story itself. The Spanish did not immediately seek to initiate a trans-Atlantic slave trade upon the beginning of their colonization effort. It was much simpler to use the local population of indigenous people as a labor source and they did so on a massive scale.
Several major problem emerged with this practice, however. First, the indigenous peoples were dying by the millions as a result of European epidemics. Enslaved Natives were even more exposed to disease than others and died at younger ages and in higher numbers. Second, most Native slaves were procured from the immediate hinterlands of the colonial outposts. They knew the land better than the Europeans and, assuming they had not been moved to a different location, could make their escape. But disease was likely a bigger factor, as it produced the myth that Native people were simply unfit for slave labor because they were too frail. A Dominican friar, Bartolomé de las Casas, launched perhaps the most prominent anti-slavery pamphlet of the 16th century and argued that, in addition to their deplorable treatment by the Spanish, the Natives were too frail for slavery. His solution was to advocate for African slavery instead, as he saw they were less vulnerable to disease and thus more fit for the labor. To his credit, he later came to regret this stance and rejected slavery as inherently evil.
Using Africans as enslaved labor was advantageous for the Europeans for a number of reasons. Firstly, they were more resistant to Old World diseases and died in far fewer numbers than the Natives (though they and Europeans all still died in droves from disease). It was also easier to control the Africans, but not for any innate racial/cultural reasons as the Europeans often proclaimed. The Africans were far away from home and had nowhere to run. They were usually separated from family and friends on the boat and surrounded by strangers who rarely spoke the same language as them. Isolated thusly, Africans had fewer avenues for freedom or resistance than Native Americans, they they both still committed suicide as a form of resistance or, more likely, a form of despair.
This pattern repeated itself in North America as well. Native slaves, usually procured by other tribes in the various wars that erupted throughout the continent as a result of colonization, were used alongside African slaves in New England, New York, Virginia, New France, Louisiana, and South Carolina in the 17th century. But these very same slave trading practices, in addition to the continual encroachment and exploitation from the Europeans, provoked violent responses from the Native peoples. These wars included King Philip's War, the Anglo-Wabanaki Wars, and the Yamasee War, all of which wrought terrible destruction in the colonial frontier and claimed thousands of European lives.
Europeans began to reject Native slavery in the late 17th century for two main reasons. Those tribes with whom they sought alliance were often deeply disturbed by the practice, which sometimes saw their close kin enslaved by Europeans, and the colonists outlawed the practice to in an effort to preserve alliance and prevent slave raiding from further disrupting Native societies. The other reason is that, as a result of the Indian Wars and epidemic diseases, Europeans came to see Natives as both too frail and too uncontrollable for slavery. They turned to African slavery for the same reasons I outlined above.
The idea that Natives were too frail and wild for slavery was interpreted through the years as them being unfit for European "civilization" in general. Europeans framed their history in such a way that juxtaposed their advance with Native decline. The tide of colonization in their narratives supposedly swept the Natives away, leaving no room for them to exist within colonial society in any form, including slavery. Natives continue to live within colonial space, but the historical narratives that begin as early as the late 17th century asserted their total destruction and erased them from the record. Take the Narragansett for example: they fought the English in King Philip's War in 1675 and, as the story goals, were totally annihilated and erased from the world. This story prevented the descendants of the survivors, who never left New England despite the colonial narrative, from receiving federal recognition until the 1970s.
Sources:
The Indian Slave Trade: the Rise of the English Empire in the American South, 1670-1717, by Alan Gallay
Indian Slavery in Colonial America, edited by Alan Gallay
Bonds of Alliance, by Brett Rushforth
Memory Lands, by Christine DeLucia