I sort of had an idea in which a world with 2-3 major continents are separated from a much larger landmass(es?) by a vast ocean. The "Old World" would be in a time period analogous to our Victorian Era, perhaps a bit before. Would it be plausible (hell, even possible) for the "New World" analogue to have developed a geo-political and maybe even cultural landscape not unlike Europe in the 1600s? If left to their own devices, of course.
When we're talking about different civilizations and their relative levels of 'Advancement' we need to be extremely careful since this is a topic is often rife with essentialism, cultural and even racial superiority. The world isn't like a game of Age of Empires transitioning from the 'Dark' ages to the 'Advanced' ages neatly, people around the world do their best to live most comfortably in the world around them, that sometimes mean developments that seem really strange or counter intuitive to those from outside the culture, especially people from the West who often see technological sophistication as the ultimate expression of an advanced society. An example would be the Aboriginal inhabitants of Tasmania, who appear to have largely abandoned some of the technology used by people on the mainland, such as fire or bone fishhooks, simply because the environment around them didn't require them.
Additionally what we think of as 'Advanced technology' or what have you can be deceptive, when the Aztecs were fighting the Spanish its often assumed that the metal armor and firearms gave the Spanish soldiers a distinct advantage over the Natives, who used a kind of hardened cotton, but the truth is far more complex. Full Metal armor was not all that useful in the hot, humid central Mexican valley or Yucatan peninsula, it was heavy and would become extremely hot, in fact most regular Spanish soldiers actually used leather armor and it was quite common for the Spanish to grab the well made Aztec cotton armor from their dead enemies if they could. Likewise firearms at that point were most useful as a shock weapon against people who had never seen them before, most Native American groups that came to understand what they really were also came to realize that 16-18th century firearms aren't all that good, especially in the low numbers that they would have appeared in the Americas, they were extremely inaccurate, very slow to fire and gave away the shooters position in a huge cloud of smoke, it wasn't until repeating firearms were invented that their advantages over bows and arrows became unstoppable, there's a reason the Model 1873 is called 'The gun that won the west'.
Anyway, when we look at the Americas its not really that hard to find cultures that were shockingly sophisticated when you consider the wider environment and how their technology compared to west, my favorite example is the Incas, who were able to weld together one of the most centralized states that existed up until then, in the oppressive environment of the Andean mountains, creating huge public works in irrigation and road-building (on vastly larger scale than anything in contemporary Europe) and built large cities, with no metal tools, wheels or writing (although some people argue the Quipu was a form of Writing totally different from any other on the planet)
But despite all this we can't deny the fact that by and large the Native American cultures were defeated and crushed by the European Arrivals, why did this happen? Today a lot of people argue that it was due to disease, rather than technology, the Native Americans proved exceptionally vulnerable to the deprivations of Smallpox, Measles, Plague and lots more, this is an extremely controversial topic but figures for the total population loss of the Native Americans, almost all due to these epidemics, suggest somewhere in the region of 70%-90% of the original population of the Americas died between 1492-1700. I would argue that this is because if you look at American societies prior to Columbus in comparison with Eurasian societies they generally had far less domesticated animals, you only really had, throughout both continents, Turkeys, Dogs, Guinea Pigs, Llamas and maybe the Muscovy Duck. In comparison Asia and Europe had Horses, Cows, Dogs, Sheep, Goats, Buffalo, Pigs, Chickens, two species of Camels, Pigeons, Cats, Yak, Donkeys and others. This meant two things, first, people in the old world lived in close proximity to their more numerous animals, which meant it was likely that disease could jump between them, indeed some of the worst diseases in history, and the diseases that worst effected the Americans, such as Measles, various kinds of influenza, tuberculosis and others, came from domestic animals or could be carried by them. Herds of cows or what have you could act as a reservoir to hold these diseases between outbreaks.
Next, the existence of Horses, Donkeys, ox, and camels, big pack animals and transport animals, meant it was far easier for Old Worlders to travel vast distances compared to Americas and to move goods over that distance, in america you could only move as far as your legs could carry you, and carry as much as your back could manage, a horse is much better for travelling huge distances, while an ox or camel can transport far more goods, whole societies were built off of their dependence on Horses to survive in huge Eurasian Steppe (i.e. the Mongols) where they could range hundreds of miles every year. This meant that lines of communication and trade were far more restricted in America than the Old world, which meant much less technological and cultural exchange, while Europe eventually got number systems and Mathematical concepts from India or Gunpowder from China the Incas and Aztecs don't seem to have been aware of each others existence and had no exchange or trade or ideas, American cultures lived within a small cultural area. This also, perhaps just as importantly, meant that diseases could spread much more rapidly throughout the old world, the Black Death is the perfect example, it started far to the east, was transported west by a Mongol army to the Crimea and transported to Europe by naval trade routes along the Mediterranean.
This all meant that Europeans had a long history dealing with a large amount of epidemic diseases that Native Americans could not imagine, they had time to build up natural resistance while and their contact with America meant the Natives, who had no resistance, died in droves to a barrage of hundreds of years worth of diseases that had rampaged through Eurasia for centuries.
So if you want to create a world of Natives who can stand against the onslaught of the old world I'd suggest that they at least have access to some of the major domestic animals, especially pack and transport animals which will make it easier to spread new ideas and Technology, while also allowing them to hit back with some of their own diseases and resist incoming ones.
For starters, 'on par'--in the real world--is a misleading concept when comparing the societies. The America's had elaborate advanced societies as well. They were just different.
The other big thing to note is that 1600's Europe was a society that formed from a long history of interacting with many other very diverse societies.
So had Europe been left to their own devices, they never would have achieved the 1600's Europe that came to be.
As such, hypothetically, it's doubtful that more isolated societies would advance the same way as societies with a lot of interaction with other societies. Even if both societies had access to the same types of diverse other societies, advancements would likely not mirror each other due to pure randomness. The reason we have different cultures is because of that.
The answer is almost impossible to determine, since there were a great many environmental and cultural factors in play. Many of the peoples in North America lived in a neolithic technical and cultural environment, while Europeans and Asians had moved to agriculture, metals and non tribal political, social and cultural structures starting @ 5000 BC. The reasons for the change are not really clear to us, and in some alternative universe an explorer from the Mohawk Confederacy could cross the Atlantic and be amazed at the neolithic peoples of the British Isles....
The answer may simply be that the Americas were much more resource rich (in easily accessible to neolithic peoples resources) than Europe or Asia, relieving them of any pressure to innovate to overcome resource bottlenecks. After all, it only took one group of people in the Old World to start the process going, and from there it spread as more and more people could observe the short and medium term positive effects of agriculture, metalworking and other innovative social and political structures being developed.
At any rate, it seems fairly certain that the peoples of the America's did have some very limited contacts with the Old World even in ancient times (ships blown off course, overly bold whalers and fishermen, the Vikings), but since these incursions would be very limited in scale and scope, they don't seem to have had much of an effect. Only large scale exploration and settlement by the Europeans starting in the 1500's seems to have been decisive, and even then the native people generally seem content to simply adapt what was immediately useful, such as firearms and horses, rather than wholesale adoption of the cultural values which were the true source of power for the Europeans.
There seems to be no evidence that the native peoples of the Americas were ever going to take that next cultural step beyond the highest levels of civilization they did attain: the Inca and Aztec empires. This is not to say they were not capable, simply that the circumstances which drove the development of European civilization to the Enlightenment were not present.
(I might note that the same circumstances were also missing in virtually every other civilization, and even in large parts of Europe itself, so the Enlightenment is also something which could have been a "what if?")
I see no reason why not. The only reason they were behind technologically is because the Americas had not been settled as long as Eurasia abd Africa. Given time they would have advanced at the same rate, remaining consistently behind Europeans until trade developed between the two, Trade would lead to ideas being exchanged.
"1491: The New World Before Columbus" is an interesting book that relates to this topic. It describes the Americas before European arrival.The Native Americans in the 15th century were pretty much on par with Europe technologically, it's just that technology is influenced by culture and environment, so their tech was different. 90% of people in both regions were peasants living in shacks, and really neither Europe nor the Americas had anything we could call advanced technology. I would say that the European advantage was that they had two things:
Large domesticated animals that let them trade and conquer over long distance, allowing for big empires, and with the Chinese for inventions (paper, gunpowder, the compass, silk, kung-fu movies, etc.)
A culture that rewarded profit (from gold, plunder, etc.), and religious conversion of those darn heathens
Give those things (and some handwavium) to the Native Americans, and contact might have been more of a meeting of equals than a mugging.