I'm pretty sure the Europeans would eventually win, not so much because they had the better technology (Sub-Saharan africa had iron, guns, and organised militaries, and the only thing that kept them safe was diseases) but because they had the more aggressive and flexible society. Writing is one invaluable asset, as is the integrative power of a religion that uses the concept of conversion and missionary work. Contract law, the idea of state and bindingly declared war and peace also help.
I think, though, that it would take an awfully long time, and most of the native population that survives the first wave of epidemics would be accomodated and integrated rather than exterminated or displaced. After all, in military termsthey aren't going to be quite as negligible. Sure, a large military force of heavy cavalry, armored infantry and archers could break almost every tribe, but most medieval militaries were small and temporary, and you still had to live alongside the Mandan or Cherokee after the heribannum expires and the knights and mercenaries all go home. Whatever tribal groups are politically dominant at the point of first contact stand a good chance of surviving as entities, though their nature will eventually change quite fundamentally (the last pagan Great Prince of Lithuania, already as feudal grandee more than a tribal warlord, married a Polish princess and converted to Christianity IIRC in 1386, founding the Jagiellon dynasty and Eastern Europe's Catholic superpower. You could well imagine a Cherokee chief or Iroquios or Mound Builder council leader playing a similar role)
And once they figure out how a horse works, the Great Plains tribes are going to make waykewl turcopoles...
(Actually I think I have the abandoned husk of a timeline with a similar premise lying around somewhere. I'll try dig it up.)