genocide: the policy of deliberately killing a nationality or ethnic group (Collins Dictionary)
Genocide is a deliberate and intentional act of fatal aggression aimed at a particular a people of a particular nationality or ethnic group. Accidental extermination because of "a single Spanish slave" carrying the smallpox contagion is not genocide though popular jargon erroneously equates accidental extermination with genocide.
This was a fascinating historical event, not in the least because the genocide that the Spanish brought to the New World was catastrophic in its ultimate consequences, but also because so much of it was committed in the second wave by accident, unintentionally causing extermination through accidental introduction of disease.
The initial conquests, like Pizarro's conquest of the Incas, came through intentional fury or aggression toward the Native peoples. In Pizarro's conquest, within 24 hours, "7,000 Inca warriors" were "slaughtered" because of the inciting event during which Atahuallpa threw a Bible to the ground. This literal genocide, as Diamond says, occurred because the Incas fell to guns, and steel.
The germs came after and were accidental, not intentional as is required in genocide, though their decimation was real (some scholars find cases of intentional slaughter of Northern Native Americans through deliberate introduction of disease committed by the English in the Northeast colonies, but documentation is still sparse).
As Jared Diamond states, "Far more Native Americans died in bed from Eurasian germs than on the battlefield from European guns and swords." This second, accidental wave of "genocide" that occurred in the Americas was mostly the result of the germs and disease that the Europeans brought with them. For example, Diamond cites the case of Cortes, who landed on the coast of Mexico with 600 men. Initially, his military expedition to defeat the Aztecs was moderately successful, with him loosing two thirds of his men, although he was able to make it to the capital city and back again in one piece. However, the Aztecs learnt very quickly how to fight the Europeans and his next foray was much more challenging.
What helped Cortes massively was the disease of smallpox. Because of the lack of exposure to this new disease, this became an epidemic that killed nearly half of the Aztecs, making this people easy to defeat. Note how disease was crucial also in the genocide of North Americans:
...North America also supported populous Indian societies in the most logical place, the Mississippi Valley, which contains som of our best farmland today. In that case, however, conquistadores contributed nothing directly to the societies' destruction; Eurasian germs, spreading in advance, did everything... These epidemics had been transmitted from coastal Indians infected by Spaniards visiting the coast. The Spaniards' microbes spread to the interior in advances of the Spaniards themselves.
Thus it can be seen that the genocide that the Spaniards committed after the initial conquest of a people was largely by accident, in that it was something that was not anticipated or expected by either the Spaniards or the indigenous population. What caused the vast majority of deaths were the microbes that the Spaniards unknowingly brought with them, which would change the face of the world through wiping out whole populations and civilisations woefully completing what was begun by the aggressive, intentional slaughter of genocide during conquest.
There are at least two ways to understand this question. Let us look at each.
One way to understand this is to say that it is asking us “how can we say that there was a genocide of Native Americans?” If this is the meaning of the question, the answer is that we can say this because so many Native Americans died. We can define genocide as the killing of a large percentage of a population of people. Because of the presence of the Spanish, a very large percentage of the natives of the land that they ruled died.
A second way to understand this is to say that it is asking us how the Spanish managed to commit genocide against the Native Americans. Here, I would interject and say that it is controversial to say that the Spanish did commit genocide. Genocide seems to refer to a deliberate attempt to kill most or all of a population. It is not absolutely clear that the Spanish tried to kill all the Native Americans in the same way that the Nazis tried to kill all the Jews or the Hutus tried to kill so many of the Tutsis in Rwanda. That said, let us look at how we can claim that the Spanish managed the genocide after the initial deliberate genocide committed at the time of conquest.
One way in which the Spanish managed to commit genocide was by introducing germs. The Native Americans had never been exposed to European diseases and had no immunity to them. This meant that “virgin field” epidemics arose, killing huge numbers of Native Americans. The other main way in which the Spanish can be said to have committed genocide was through their policies about making the Native Americans work. We can say that the Spanish committed genocide by putting the Native Americans in situations where they were being worked to death.
In these ways, we can say that the Spanish committed genocide against the Native Americans, but this remains a controversial statement.