Won't have a communicable disease? During this period? Are you serious?
Yet another gross conceptual error. It doesn't need to be smallpox or anything you perceive to be nasty. It could be something Europeans considered to be a childhood disease like the measles or the mumps. It could be the flu. It can be nearly anything because it's a virgin field.
All typhoid fever requires for transmission is infected feces and guess what is one of the last things your body does as it turns into a corpse? Typhoid killed over 5000 people in Jamestown in less than two decades during the start of the 17th Century and we can only guess what it did to the surrounding tribes.
Need another candidate? How about typhus? All that needs are lice and they're really hard to eradicate. It also "loves" crowded conditions like those in jails or ships. And guess where our earliest verified account of typhus occurred? During a Reconquista siege of a castle in Spain in 1489.
As they say, THIS.
Particularly the underlined parts.
Even if the sailors are perfectly healthy by European standards (as in, wouldn't infect any other Europeans), the normal bacteria floating around their bodies, their stuff, and their systems will be problematic.
The question is not if something would spread, but what would spread.