We can clearly look back now and see that European diseases like smallpox killed millions of Native Americans. Did the explorers realize that this was occurring, or did they think it was mostly coincidence?
Also an unrelated question, what is the 'correct' term for referring to the people who inhabited North America (including the Caribbean) prior to the arrival of Europeans? I've heard "natives", "native Americans" and "Indians" from professors. Which is considered correct?
For the second part, there is not really a consensus as to what is the "correct" term, since some people (being individuals and not an abstract concept) might prefer one term over another. You can probably never go wrong with referring to a group by their specific name, but more general terms like indigenous/native can work depending on context. "Indian/Indio" tends to be either accepted or rejected. Certain nomenclature usage is dependent on political boundaries, like Native American/First Nation in the US/Canada. Finally, groups in the Arctic tend to have their own nuances in describing them as a group. In historical context, I prefer to just use "Americans," because that's what they are. The easiest way to figure out what the right term to use in an academic context is to familiarize yourself with the current literature on the topic. The easiest way to familiarize yourself with what someone wants to be called is listen to what they call themselves. The easiest way to be mistaken for a time-traveling academic from the late 70s/early 80s is to use the term "Amerindian."
On to your actual question. If we look at Mesoamerica, we can see again and again the idea that the Conquistadors, and the friars and settlers who came after them, did not have anything resembling a modern understanding of disease. When epidemics occurred, they were written about as though they were natural phenomena like the weather. Take, for instance, this passage from Acosta's Natural and Moral History of the Indies, published in 1590 (Lopez-Morillas 2002 trans.):
There is a large Indian population in all the mountain regions; they maintain their numbers today, and some even say that the Indians are increasing, except that work in the mines kills many of them, and some common diseases (such as the fever called cocoliste in New Spain) have caused many deaths; but in fact it does not seem that their numbers are decreasing.
That passage is bonkers. Acosta was born in 1540, so he would have been too young to experience the first round of epidemics in the 1540s which reduced the indigenous population in Mexico by more than 50%. He was also Peru during the second round of huey cocoliztli, which further devastated the native population. In other words, he showed up after the main epidemics had already occurred. So let's ditch Acosta and look a closer source who was there during the very first epidemic of Afro-Eurasian disease in the American, Cortes (3rd Letter).
The natives had asked me to go [to Cholula], since many of their chief men had died of the smallpox which rages in these lands as it does in the Islands...
Oh, OK. What else does he have to say?
On the following day all the chieftains of the city and the province came to me saying that [Maxixicatzin], the principal ruler of them all, had died of the smallpox...
There is literally no indication that Cortes has any epidemiological concept of disease. It is, to him, something that just happens. Smallpox was endemic in Europe, so why would it no be endemic to the Americas?
Let us try then, another first hand account of the 1520 Smallpox epidemic, Bernal Diaz del Castillo (True History of the Conquest of New Spain):
he Mexicans and Tezcucans were not upon the best of terms with each other; and the smallpox was raging throughout the whole of this district, so that they were not able to bring many warriors into the field.
Again, not exactly descriptive and the idea of intentionality is outside the discourse, though there is an awareness of the effect it had. We can see a clearer example of the disease schema at the time from a later passage:
In the year 1527 there appeared in the heavens a sign, which had the shape of a long sword, and seemed as if it stood between the province of Panuco and the town of Tezcuco, and remained unchanged in the heavens for the space of twenty days. The Mexicans and their papas declared it was a sign of some pestilence, and certainly a few days after the measles, and another eruptive disease, like leprosy, broke out, which was accompanied by a very nauseous smell, and carried off numbers of persons, though it did not prove so destructive as the smallpox.
There, again, is the recognition of disease as devastating, but again as an inevitability of life. This is coupled with a supernatural view of disease as well.
Cortes and Diaz del Castillo, however, were soldiers and not men of learning. Let us turn to the Court Physician of Spain, Francisco Hernandez, who was in Spain during the 1570s epidemic and see what he has to say about it:
The epidemic started in June 1576 and was still not over in January, when we wrote this description. From New Spain it invaded all the cold regions in a circle of about 400 miles, and wa ssomehwat easier on the warmer regions (that is, it attacked rather less) infecting different areas in turn, beginning in those occupied by Indian tribes, the places where Indians and Africans lived, then those with a mixed population of Indian and Spanish, and later still, those areas occupied by Africans, and now finally it is attacking the Spanish.
Hernandez goes on to posit that the indigenous diet may be a factor, based on the Hippcratic/Galenic humoral system under which he was operating. So here we have a recognition of differential morbidity, but still nothing like any sort of modern epidemiology.
Perhaps the most apropos to your question is that passage by Torquemada in Monarquia Indiana, the English translation for which I am shamelessly stealing from this paper you should also read:
In the year 1576 a great mortality and pestilence that lasted for more than a year overcame the Indians. It was so big that it ruined and destroyed almost the entire land. The place we know as New Spain was left almost empty. It was a thing of great bewildermentto see the people die. Many were dead and others almost dead, and nobody had the health or strength to help the diseased or bury the dead. In the cities and large towns, big ditches were dug, and from morning to sunset the priests did nothing else but carry the dead bodies and throw them into the ditches without any of the solemnity usually reserved for the dead, because the time did not allow otherwise.
Apocalyptic, but Torquemada had the advantage of hindsight; his work was not published until 1615.
I've said this before in posts on questions about giving out smallpox blankets, but human infectious disease do not need help to spread. That is literally what they have evolved to do. All they require is an ignorance of germ theory and sterile practices that were common place at the time. Moreover, while pathogens take dramatic center stage in the demographic collapses of the Americas, we must not forget the other important players were the exploitative and often barbaric policies Europeans enacted in the wake of epidemics. Amherst, in writing about his intention to give Americans materials he believed contiagious for smallpox, expressed his frustration in not having enough men to enact the "Spanish Method" of hunting his opponents down with dogs and horses. De Las Casas, in his Destruction of the Indies, barely mentions disease, because he has ample evidence of the brutal enslavement of the native population leading to men being mutilated and burned alive while the women simply abandoned the notion of having children. The European colonialists that arrived in the Americas had, at best, a vague notion of contagion, but they were able to exploit the results of their introduction of novel diseases to the Americas.