So this touches on the question of "why did the Americans get devastated by European diseases and not vice versa?"
The most common answer I've heard is that the Eurpeans were in close contact with large animals but the Americans weren't. How true is this? Weren't the Inca in close contact with llamas?
The quick and dirty answer is that Europeans did fall ill and die in the New World, but we may never know, with 100% certainty, if they died from New or Old World pathogens. Check out this previous answer for more info.
In addition to what /u/anthropology_nerd's comment regarding the diseases that did befall European colonists in the Americas, I'd like to point out that case attributing major diseases is almost certainly overstated. It was popularized by Jared Diamond in the late 90s and early 2000s, but more recent research is showing a more complicated picture. In this post I link to several articles concerning the origins of various major diseases. With the exception of measles, which seems to have merged within the last few thousand years from the closely related rinderpest virus that infects cattle and other hooved animals and possibly mumps which show affinities to a virus that infects both pigs and bats, few of the most deadly diseases can reliably be traced back to domesticated animals. Smallpox and tuberculosis, for example, appear to pre-date animal domestication by thousands of years.