It is generally believed that syphilis was brought back to Europe by Columbus' men. The disease was not reported on (which doesn't necessarily mean that it wasn't present but more than likely it wasn't present) in Europe before contact with the Americas.
Source: The origin and antiquity of syphilis revisited: an appraisal of Old World pre-Columbian evidence for treponemal infection
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I don't know if you would count them as "Native American" illnesses, but the climate of certain areas of the Americas was different enough from Europe that they encountered new illnesses as a result. The many varieties of "jungle fever" are just one example that most of the settlers wouldn't have immunity to, but aren't exclusively American.