This is a frequently asked question, and more can always be said, but I'll quote an earlier answer below...

There is very good reason to suspect Native Americans did have their own set of disease that Europeans were not prepared for.

The quick and dirty answer is that Europeans did often fall ill in the New World, and in many cases we assume these deaths were from diseases they encountered in the Americas. Contrary to popular opinion, the New World was not a disease-free paradise. About a year ago I wrote a post about evidence for epidemic diseases in the New World before contact. In that post I mentioned New World populations played host to a wide variety of intestinal parasites (roundworm, hookworm, whipworm, etc.), gastrointestinal diseases (Giardia, Entamoeba, and Cryptosporidium, etc.), Chagas disease, syphilis, Rocky Mountain Spotted Fever (and possibly Lyme), and tuberculosis. I also hypothesized they would be subject to occasional zoonotic events (when a non-human pathogen jumps into human hosts), just like modern populations with frequent access to wildlife/bushmeat trade. There is also reason to believe that observed epidemics that occurred after contact, like the cocoliztli (a Hanta Virus-like hemorrhagic fever) epidemics that swept through Mexico in the sixteenth century, were present, though perhaps more contained, before contact. Two cocoliztli epidemics, in 1545 and 1576, killed between 7 and 17 million people in highland Mexico, Europeans included. There is no evidence the pathogen responsible for the epidemic arrived from the Old World, but researchers suspect a massive drought altered the relationship between the murine host and humans, leading to increased chance of pathogen transmission, and a catastrophic epidemic. New research has muddied the cocolitzli argument, however, after the discovery of Salmonella enterica in contact period mass burials.

Next, when we read the accounts of early Spanish entradas in the U.S. Southeast, the authors make specific mention of crew members becoming ill weeks after their arrival in new lands. Nutritional and physiological stress from poorly planned colonization attempts likely decreased their immune defense, leaving them vulnerable to illness. Ayllón's 1526 attempt to establish a settlement on the Santee River in South Carolina ended in disaster. Of the original 600 colonists, all but 150 died from hunger and disease. Later, the 1528 Narváez entrada likewise suffered a series of unfortunate events in their attempts to find riches in Florida. 400 men landed in Tampa Bay, yet only four survived the trip to Florida. After a month of raiding Apalachee towns, members of the entrada began to fall ill. Cabeza de Vaca says

there were not horses enough to carry the sick, who went on increasing in numbers day by day... the people were unable to move forward, the greater part being ill.

Did members of Ayllón and Narváez's entrada perish from New World pathogens, or did they bring their own microbes with them, and perish as a result? We don't know for sure. The deaths began outside the incubation period for many common pathogens, giving us reason to suspect they did not bring those illnesses with them from Cuba, but rather encountered them from the neighboring maize-based agricultural populations like the Apalachee. Similar European mortality events are noted in Jamestown, where of the > 3,500 who arrived from 1617-1622, only 1,240 were alive in 1622. The chief cause of death was endemic illness, and the term "seasoning" was commonly used to describe the disease transition new immigrants needed to go through before their survival was more assured. We don't know for sure if the seasoning illnesses were infections brought from Europe, or if they included pathogens encountered for the first time in the New World.

To sum up, the popular assumption that Europeans did not encounter any new pathogens in the New World looks to be wrong. As we dive into the primary sources we find abundant evidence of infections, but it will always be a little difficult to determine, with 100% certainty, that those illnesses were from New World pathogens alone.