Well, the question wasn't whether there could be an Americas-wide spread of Mesoamerican/Peruvian level civilization but what would be the result in the 16th century if there was.
Given the range of cultural expressions and technologies among the various Mesoamerican cultures and the various Peruvian cultures, Columbus and later explorers would immediately encouter a continentent divided among hundreds of small empires, city states, and kingdoms, all of which would possess complex and heirarchical social structures with hereditary nobilities, efficient maize, corn, and bean agriculture, intricate polytheistic pantheons, and varying degrees of urbanism. Some would possess complex written languages based on the syllabic and logographic Mayan script, others would use administrative record-keeping devices like the quipu, others the pictographic Mexican systems. Bronze metallurgy might be wide spread and diffusion if ideas between Peru and Mexico would probably have resulted in wider use of domestic draft animals. I question the idea of the "Mexican wheel", although it is remotely possible the availability of draft animals in Mexico might have created a reason for the toy to be given are real function.
A few states, most likely in Mexico and northern South America would probably be the most complex and static, while "barbarian" states in north america and places like Argentina near the margins of the core area might be more dynamic, flexible, and adaptable - having presumably been "civilized" for a shorter time.
This would certainly slow down, but perhaps not stop, the eventual European colonization of the new world. However, it might well assume the form it did in India, the near east, and parts of Asia - with Europeans assuming dominance over the American states by usurping local leaders, but with native languages and cultural traditions largely surviving intact. If (and it's a big if) the effects of disease could be curtailed, Europeans would probably not not become the dominant population group anywhere in the new world except perhaps for the extreme north and south where tropical maize/bean/squash agriculture might not be as effective as European wheat-based cereals.
Even as subordinant peoples, the local civilizations would quickly adopt elements of European technology such as iron, cattle, horses, effective wheeled vehicles, and other attributes of a medieval iron age society.
It's hard to say about religion. In some areas I suppose it is possible that native paganism might survive as Hinduism did, or synchetically mix with Christianity, but if these societies practiced cannibalism and human sacrifice as in OTL, I can't help but believe Europeans would attempt to overthrow the native faiths one way or the other. European colonialists could cooexist with subordinant Muslims or Hindus, but I doubt of they would tolerate Native priests dancing around in the flayed skins of too many virgins for too long without getting positively Spanishly Inquisitional.