Most likely not any time soon.
If the original post refers to just the Spanish not showing up (with lots of other European powers soon attempting to conquer the region) nothing in regard to your question would have changed.
In case of a significantly different timeline (with no European exploration/conquest) of the Americas and at least several centuries of "undisturbed" further development of the Pre-Columbian cultures: Yes, some later post-Aztec successor state might have expanded into that region.
Let me explain, how:
The problem with alternate history is, as you know, that none of this did happen and therefore we lack hard evidence on what would have happened. It is akin to conduct a study based on a single observation (for any historical event); we do not know the ensemble mean, we do not know the variance, we do not know anything else. How likely is it that something specific would have happened.
To illustrate this: How likely was it that Alexander the Great conquered the greatest empire of the age with an unruly lot of modestly trained peasant-soldiers? Hence: Imagine a strong Aztec leader who, for whatever unlikely reason wanted to conquer the area of modern northern Mexico, Arizona, and New Mexico (perhaps because Aztlan, the mythical Aztec homeland, was said to be located there), perhaps she would have.
What we are left with is to speculate about the likelihood of certain events based on analogy to other events in another time and place in world history and a bit of logical deduction. Of course, you know that; I am just trying to put my argument on solid foundations.
So let us see what we have.
The Spanish do not show up
This is very good news for every Pre-Columbian culture. Not because of conquest, not because of rape, murder, and imposition of nonsensical health care practices (apply bloodletting liberally but never ever take a bath). Rather because of epidemics that historically caused the death of a significant part of the population of all pre-columbian societies - as far as I know, it is not clear, how many, but it may have been around 90%. Consider the Sadlermiut Inuit as a well-documented modern example with 5 survivors out of a population of 70 in 1902/03 CE. Perhaps the death rate was somewhat lower for the larger Pre-Columbian societies, but still it would have been incredibly large and would have caused devastation also on social, economic and technological levels. It perhaps caused the collapse of the Mound builder culture in the plains region of what is today the US; basically east of the region and societies (Hopi, ...) your question referred to.
- No devastating epidemics in any of the Pre-Columbian societies
- The Mound builder culture survives.
- Perhaps remnants of an agricultural Anasazi(?) / Hopi culture in the area of interest to the present question also survive (though it had been in decline caused by significant climate changes in the area from 1200 CE)
- Many Pre-Columbian high cultures continue to develop "undisturbed".
But what does that mean?
"Undisturbed" development of Pre-Columbian cultures
The Pre-Columbian high-cultures of the time (Aztecs, late-period Maya, Inca) were at about late stone-age or early bronze-age stage. That is about the stage of development of the ancient Egyptian Old Kingdom (i.e. the pyramid builders) at 2000-2500 BCE, the contemporary Mesopotamian cultures (Akkad), the slightly later Harappan, Minoan, ancient Chinese cultures. The time difference is explained (convincingly IMO) by Jared Diamond by a mixture of factors including the size of areas with the same climate (large in the old world with vast east-west extent, small in the Americas which mainly follow a north-south orientation), the lack of domesticable large animals in the Americas, the and similar lack of equally favorable domesticable food crops in the Americas.
Now, the question would be what reason for the Spanish not conquering Mexico the original post actually intends:
- did the "old world" civilizations never developed to the extent they did?
- did the just Spanish not show up ... but instead the Portuguese, the French, the English, the Dutch and the Danish did? ... I am going to exclude this possibility, as in this case it would not have made much of a difference. The Aztecs would have been conquered a few years later by another band of European misfits and the leader's name would not have been Cortes, but beyond that everything would have been very much the same.)
- did the European civilizations develop but without developing seafaring technology or interest in going to the Americas (much the same way as China in the same period; they did have ships, they did send out their navy - just not across the Pacific ocean?
In the last case ("old world" civilizations without exploring the Americas at the time), Pre-Columbian civilizations would perhaps have enjoyed a few hundred years more without "old world" interference and "old world" epidemics before the clash would finally have happened. It is difficult to speculate, how many hundred years exactly.
In the first case (no "old world" civilizations) they would have been the undisputed technological leaders, would have eventually discovered, exploited and conquered the "old world" thousands (?) of years later.
How would the Aztec civilization have developed?
The Aztecs were not the first empire in the region, they were preceded notably by the Toltecs that basically governed much the same area for centuries before the rise of the Aztecs; culture, technology, and civilization only developed slowly (much the same way as in "old world" civilizations at similar stages of development). It is unlikely that the Aztecs would have been able to significantly expand their empire without major technological and social changes. They were surrounded by formidable opponents who they continuously fought. There is also a technological limit to the geographical extent of military campaigning; in most regions campaigning all year round would have been impractical and bad weather often led to significant losses to ancient civilizations armies. They would have had to care for supply lines, would have had to set up secure winter (or rain season) camps, etc. Both the Aztecs and the ancient near eastern bronze age civilizations (the latter up to about the year 800 BCE?) mostly conducted short military campaigns lasting one season; rulers would try to subdue opponents in a particular geographic region and proceed to deal with the next region the next year.
From about 2000 BCE "old world" empires further had the benefit of using horses (and other large animals) to increase the reach of their campaigns. Cavalry or chariots move faster but especially useful are beasts of burden; the Aztecs did not have those; perhaps, given a few hundreds or thousands of years more they might have acquired llamas from South America. They also did not have the wheel (so, no carts), not so much because they did not understand technology but rather because cart transportation only makes sense if you have a road system that is technologically advanced enough to facilitate use by carts.
Could they have eventually developed a road system, acquired llamas or trained other beasts of burden (good luck with training that buffalo ;))? Probably yes, given enough time. But you would have to expect this to take considerably longer than it did in the "old world" as a result of the geographical arrangement of the continent vertically to the climate zones.
Would it have been the Aztecs that would eventually have made these magnificent technological advances? Probably not, as empires throughout history typically did not last more than a few hundred years (with the notable exceptions of China and ancient Egypt - perhaps these much later post-Aztec conquerors would have seem themselves in the tradition of the Aztec empire in the way medieval Chinese saw themselves in the tradition of the Yellow emperor, but they would have been very different from the Aztecs we know.)
- Continued slow development of civilizations in the geographical area of central Mexico (Aztecs), Yucatan (Maya), the great plains (Mound builder), Andes (Inca, Muisca).
- slow progress of domestication, distribution of technologies and domesticated species, development of infrastructure (roads, ports), cultural exchange.
- all this would have been slower than in the "old world". For getting from early bronze age technology to 16th century technology, the "old world" required 4000 years. It would have taken the "new world" significantly longer.
Given technological improvements, would the (post-)Aztecs have explored, attacked, and conquered the post-Anasazi, post-Hopi etc.?
Why would they want to do that? The Aztec empire already expanded into really far-away (given the technology level) areas. However, they were located South of their core lands, not North. (The farthest point being, afaik, Xoconochco.) This is precisely because the economic and cultural focus of their world was in the area stretching from their core country southward to the late-period Maya civilization in Yucatan, Guatemala, Belize, ... They had (loose) cultural ties and economic exchange with this region; there was very little for them both culturally and economically in northern Mexico and beyond. (Except, notably, that their ancestral homeland, Aztlan, was supposedly located there.)
But with social, economic, and technological development, this might have changed for 3 reasons I can think of:
- new technologies require new resources. There seem to be significant mineral deposits in the Rocky Mountains; they may have started to exploit those.
- with the developing Mound builder culture, the region of the modern US would have become an interesting target for conquest by (post-)Aztec rulers; this would have required control of the regions in between (or such control would at least been beneficial for any military efforts of that type).
- It is possible that tribes in modern Northern Mexico and the South-Western US repeatedly migrated into central Mexico, attacked the states there, and tried to conquer it. Notably, the Aztecs traced their origin to a mythical land in the North, Aztlan; the Aztec language (shared by other groups of the area) also is part of the Uto-Aztecan language family to which many of the indigenous languages of the Western and South-Western US (including Hopi) belong; this suggests immigration from that area. If that happens repeatedly to an advanced civilization (as it did to China and to the Roman Empire), the respective authorities will eventually attempt to deal with it, typically by building a wall (like this and this) and by occasionally sending armies to rough the potentially threatening tribes up (like so and so).
So, yes, much later, interference in the area by a more advanced empire from central Mexico (after centuries or millenia of development undisturbed by European or other "old world" forces) would be conceivable.