While your scenario is highly especific and I find implausible, I'll like to state my opinion before you get flooded in cries of 'ASB!'
I don't think Native Americans sailing to Europe is an impossible scenario. Most people discount their naval skills. The Caribbean peoples had large canoes that they used to trade between islands, and the Patagonians are well known for their exploration of lands upriver; some people from Chaco, in central South America are descended from Patagonian peoples. And just in case anyone still thinks the Native Americans were all nomads with the exception of the Inca and Aztecs, they were not. Many of them were organized in cities and complex societies, and the current estimates for population before the plagues and conquest are higher than ever.
However, the Europeans had huge advantages compared to the Native Americans, most prominently of them sails, compass and navigation instruments. It is crazyness to compare the native canoes to a caravel (except for size; some were longer than Columbus' caravels). However, what if some natives have developed seafaring to the extent of the Old World civilizations? Perhaps they discovered the sail for better fishing voyages, and some enterpeneurs use them to trade across the Atlantic coast. They could look, maybe like Trirremes (after all, it would combine the canoe with sails). Then, maybe some of those 'Trirremes' get lost outside the coast of Brazil and contacts Africa. Or some fishermen going north make contact with Viking expeditions. I doubt however, that a 'Sunset Invasion' is plausible at all. The Old World will not fall to disease as the Americas, and the Americans are still at a technological disvantage.
It depends on how does this happen, you don't need a huge conquering empire. The Caribbean peoples could create a trading empire across the Atlantic coast, or maybe even the Amazonic peoples (that we know so little about, but they had complex societies and were expert fishermen). It is highly improbable, but NOT impossible, and possible even without European influence.