I've always thought the term "castle" referred to a very specific form of fortress that was the permanent residence of the regional ruler, lord, or warlord that could also be used as a temporary refuge for his subjects in time of siege. Do star-design fortresses built by Britain, France, Spain, or the US really qualify as castles? There are examples of these in many areas of North America, but to my knowledge their purpose is mainly military, with a lesser emphasis on trade with and protection from indigenous peoples or other European states? For old world examples, were Hadrian's wall or the Great Wall of China built as "castles" or as the specialized border fortifications of organized states?
I would imagine that western stockade forts come closer to the concept of "castle", but actually the walled towns and elevated earthworks of pre-contact and early contact Mississippian cultures come closest. These structures often enclosed only the civic and ceremonial centers of these towns (ie: where the high status ruler, his family, his religious and civic officials and retainers lived) while presumably the people who lived in farmsteads and hamlets immediately outside of the stockade could take refuge in them during war.
Without assuming a much earlier European discovery and settlement of the Americas during the early middle ages (pretty unlikely), I'd look to native cultures as providing the likeliest source for the evolution of something like the European castle in America.