Well, to start with, this hypothetical isn’t wildly different from what did happen. The Spanish conquistadors did carve out enormous personal holdings where their word and whims were law, and from which they freely raised armies to conquer more and more of what they called New Spain. They didn’t need to declare political independence from Spain because King Carlos was too far away to interfere in their day-to-day operations, and maintaining their connections with Spain was crucial because the riches of the Aztec empire weren’t worth much without the markets of Europe to sell to. Hernan Cortes was a bit of a bandit; he was only assigned to explore the coast, and when a larger Spanish army was sent to arrest him and bring him back to Cuba, he overwhelmed them, absorbed their army, and continued conquering the Aztecs. Besides, as ambitious as these men were, they still saw themselves as Spaniards, and while they wanted to become wealthy, powerful and aristocratic Spaniards, the dominant governmental philosophy of the age is still the “Divine Right of Kings,” where rebellion (revolution isn’t really a concept yet) isn’t just a crime, it’s a sin! The conquistadors continued to not only send their kids back to Spain for schooling for the next two hundred years, they even sent their pregnant wives back to Spain to give birth there because the social status of a Gauchopin (a Spanish immigrant, instead of a colonial) was so much greater. The Americas were a fascinating places where a man could make a tremendous fortune, but they were cultural backwaters, and the focus of these mens’ minds was still Europe. The conquistadors might or might not be able to bootstrap themselves up to the current standard of technology, but some expensive and vital items like cannons would be very tough to produce in the New World. With the active opposition of the wealthy and fast-growing Spanish colonies in the Caribbean (now almost thirty years old), it would be difficult to get more horses, men, and other essentials. And while they might have been able to defeat any native armies, they still had legitimate worries about the other major seafaring power- the Portuguese, who were also Spain’s main competitor in the sugar trade.
If they had set themselves up as opponents of Spain, they’d suddenly find themselves cut off from the continent by the largest, wealthiest, and most technologically advanced navy in Europe (remember, the Spanish Armada wouldn’t be defeated for another eighty years). They’d own lands that produced goods of tremendous value, but they’d have nowhere to sell those goods. The Americas were undergoing a series of plagues of a scope that’s difficult for us to comprehend; some estimates run as high as 95% of the pre-Columbian population. I think that number’s a little high, but we can safely assume that absent a growing population of Spanish immigrants and African slaves, the value of the plague-devastated Aztec empire is pretty minimal. Tenochtitlan had been reduced to rubble by Cortes as he captured it.
Practical upshot? I think that Cortes would find it pretty damn difficult to rule for long in Mexico with the active opposition of the Spanish, and the loyal Spanish in the Caribbean would have tremendous incentive to invade and take over. The natives were dying off in droves. The only arch-rival of Spain in a position to interfere are the Portuguese, who didn’t know what was going on in the Caribbean at the time; news about colonies were state secrets comparable to nuclear bomb design today. England had no New World holdings at all and wouldn’t for almost a hundred years, and the prevailing wind patterns in the Atlantic meant that any English fleets would have to sail right past Spain and Portugal to get to the Caribbean.