I would say, likely never, barring outside influence. Even when you look at Europe, it was a near run thing, more a matter of luck at times than anything, and Europe had lots of advantages.
In Guns, Germs and Steel, Jarred Diamond argue that the Americas would always be slow to develop but the primary orientation of the continents were North-South instead East-West as in Eurasian (Africa has the same problem.)
This causes trouble because biomes are largely defined by latitude, which defines their temperature. Closer to the equator, hotter, closer to polls colder. That means that as you move East-West, the temperatures stay more constant than they do moving North-South.
Plants and animals can migrate or be traded East-West but far less easily North-South. Traveling East-West, you just need one suit of clothes. Traveling North-South you need several, one for each clime you will cross.
In civilization and technology building, the key factors are population size and trade. Populations tend to stick to biomes and grow East-West so if you have a primary North-South geography, you see a lot of smaller chopped up cultures and civilizations that are relatively isolated.
Population is necessary just for the surplus material a large number of people can produce and just for all the brain power it provides. Large population can also more easily back up information and survive cataclysms.
Trade is necessary to increasing technology because no place on earth has enough resources in a radius of 100-200 miles to support a Renaissance technology level. Also, trade is necessary to prevent civilization from collapsing from famines.
If your primary travel corridors run North-South, population gets chopped up into small cells, the spread of useful plants and animals is very slow and general travel and commerce, much more difficult.
The other problem with Americas is that they do not have native animals that can be domesticated, especially beast of burden. Diamond spent a lot of time pointing out that of the millions of mammal species only about a dozen have been domesticated. Domestication requires certain behavioral traits that not all animals possess. Attempts to domesticate Zebras for example, have been attempted by Islamic and then later Europeans (Zebras are immune to the tsetse that wipes out horses in Africa) but the animals are just too belligerent and easily startled.
In the Americas, the only animal that could even be considered for a beast of burden would be the buffalo, and even after a couple of centuries of people trying to raise them, they are still not domesticated just in terms of keeping them for meat and skins. Trying to yoke one is out of the question.
I think that culture in the Americas would cause a serious block. I would argue that the means of organization are a type of technology themselves, one that allowed them to accomplish more than any other cultures ever did with the same available resources. But they also created a serious early adopter trap that would have prevented further development.
The conditions that lead to the flourishing of technology, art and science are not natural and not stable. The progress of humanity is a story of this or that small polity suddenly and usually by chance, being forced to adopt a more egalitarian and merit driven society. Then they have a run of 200 years or so, and then fall back into a static, hierarchal, parton-client system where birth and political connections mean far more than merit and class mobility freezes. Then the locus of advancement jumps somewhere else. Human progress is more about following a golden thread from small pocket to small pocket all over the world for centuries.
Key to such development is escaping centralized authority and control with a subsequent increase in individual freedom and merit reward. Empires strangle off development fantastically. You can see this in the fall of the Islamic world after the Ottoman Empire absorbed most of it. A dynamic civilization slid into stagnation because Empires seek stability more than anything else and progress does not bring stability.
But the cultures of the Americas could not easily dispense with centralized authority. With only stone and wood tools to work with the high civilizations of the First Americas-peoples relied on an incredible level of social and task organization ito marshal vast amounts of skilled human labor to overcome their lack of metal tools and draft animals. They shaped stone by pounding it, not chiseling. They moved stone by huge teams of men, not by yoking oxen and so one.
While powerful, reliance on organization made the societies fragile. In Meso-America, almost any kind of social disruption caused city states to collapse economically and for their people to abandon the city, sometimes for decades or centuries.
The kind of wrenching social change in Europe that drove the the Renaissance would have simply destroyed the civilizations of Americans.
So, I think the answer would be "never" but if they did, it would take thousands of years.
