Native American peoples had extremely low genetic diversity. There was a genetic bottleneck caused by their immigration over a narrow land bridge which was later submerged. Their Asian ancestors were already less genetically diverse than Europeans. (However, Asians back then were a bit more diverse, including groups more closely related to Caucasians than to the more familiar east Asians. Native Americans do carry genes from those lineages too.) European genetic diversity is far less than African genetic diversity, but it has still been estimated that any two native Americans from completely different tribes were more genetically similar than the average first cousins of European descent.
Native Americans were so in-bred that it is surprising they they did not suffer from a great many genetic disorders.
Genetic homogeneity means that there is unlikely to be much resistance to pandemics. If a diseases moves through the population and kills too quickly, then there may not be enough carriers left to spread it further.
In much of the New World human population density was relatively low. Diseases proliferate better with dense populations, where they can move from host to host quicker.
There were some densely populated cities in Central America, but they had remarkably advanced sanitation systems. The fall of Tenochtitlan did lead to a great many Spaniards dying of a deadly native disease. However, the natives died from it too. Having a quality sewer system prevented them from developing an immunity to it, and losing that sewer left them quite vulnerable.
In Europe, Asia, and Africa, it was quite common for humans to live in close quarters with their livestock. Most major human diseases started in other animals and spread from this contact. It was also common for Eurasians to milk their cattle, which requires close contact with the delicate mammary tissues through which pathogens can travel easily.
There were some domesticated animals in the new world, like turkeys, llamas, alpacas, and guinea pigs. Humans did not live in close contact with these animals though, or rely on their milk.
I've read that the natives' genomes were actually better suited for fighting off the native parasites, but such parasites did not cause easily communicable diseases.