Note: I'm assuming by America you mean North America.

There's actually a lot of debate over what constitutes a castle, which contributes to some of the confusion as to whether a particular building is a castle or not, and conversely whether said building constitutes a castle in America or not. Because to some people, a castle is any kind of (preferably stone) fortification. Whereas to others it's a specific kind of private fortified residence.

I view "castles" as belonging to one leg of a triangle matrix of fortifications consisting of public, private, and military (as a halfway between public and private).

A public fortification would be a walled town or city. A military fortification would be, well, a fort. A private fortification (one intended for a single individual or a single family), well this is what's traditionally known as a "castle" in Europe.

But keep in mind these definitions are not hard and fast. For example there are many ex-military forts that were turned into or simply called castles, like for example Portchester Castle in England. It was originally built as a Roman military fortification, but then granted to an individual noble by William the Conqueror.

There are "refuge castles" which serve as defensive strongholds for a local population, but aren't the same as a walled town or a military camp.

The great Ring of the Avars fortification that Charlemagne captured comes to mind as a fortification that's not quite elite residence, and not quite military camp, and not quite town (because it possibly didn't have permanent occupation).

My bottom line being, the closer a fortification hews to toward the triangle leg of private fortified residence, the more likely it is a castle.

Back to America, keep in mind the "fortification" need not be functional, it can be an architectural affectation. This being the case in places like Hearst Castle, which serves no actual military function, but is constructed in somewhat of a pre-modern military style, but IS a private residence.

So the answer to your question, would depend upon what you consider a castle. If you consider a castle a private stone-fortified residence in a then contemporary pre/early modern style intended also to function as a military stronghold of that period, then to the best of my knowledge, no, such castles don't exist in America. Simply because such strongholds were intended in Europe as the residence of nobles to secure their local power of the land which was under their domain, and the organization of the land settlement in America by the immigrants meant there was no such singular local privatized power (in possible conflict with other local privatized powers) to need one. Fortifications tended to be public or military, i.e. walled towns or forts.

The gunpowder issue is only a problem if you see the definition of a castle as being required to be tied to being a functional fortification rather than having decorative fortifications. Castles of the latter type were still being built for residential purposes well past the demise of their functional predecessors.

But if you consider a castle any pre/early modern stone fortification, or you consider the private residence of a member of the elite built in a pre/early modern stone fortification style, then yes, they do exist, and there are quite a few of them in America.

I don't know enough about Latin America to provide an answer there. Perhaps someone else could shed some light on that region, considering that they had longer interaction between a conquered population and a european elite, which might have made it possible for private fortified residences.

In short, yes.

Gunpowder was common, though not advanced when america was settled.

Cannons greatly reduced the defensive value of a castle. Also the labor required to make one compared to a log pallisade, which was nearly as effective made it an obvious choice.

We kindof had castles, we had forts. Built of earth, timber, bricks and old enough settlements forts were stone.