There are a few types of castles to consider. Very broadly:
medieval fortifications, designed before guns
artillery ("star") forts, designed to defend against guns
revival castles
You won't find many medieval fortifications in America for the simple reason that Europeans did not begin building western-style fortifications on the west side of the Atlantic until the 16th Century, after medieval fortifications were obsolete. The Incas, Aztecs, and Mayans did build stone fortifications and city defensive systems, which are analogous to castles or citadels. Machhu Picchu is a famous example. The Pyramid el Castillo in Tulum looks quite castle-like, and had very castle-like walls and gates.
The Spanish began building New World fortifications in the 16th Century, so you can find some castle-like structures dating from that era. Morro Castle in Havana is a good example. As for a bona-fide castle that was the residence of a king, check out Chapultepec Castle in Mexico City, which was built in the 18th Century. A few of these Spanish castles are now in American territory, such as Castillo San Felipe del Morro in Puerto Rico.
Artillery forts, usually built in star-like patterns, took over from medieval fortification patterns in the 16th Century and continued right through the 19th. Most American stone forts follow this pattern, and there are many, many examples. I'll just point you to some Wikipedia listings to give you an idea of the number: United States, Canada, and Cuba all have numerous examples.
Revival castles were mostly built during the neogothic and medieval revival periods in the 19th to early-20th Centuries. These were actually super-popular in "gilded age" America, due to their ostentatiousness which was suited to the huge industrial fortunes that were being amassed. They were not built as fortifications, but as palatial residences, institutional buildings, or hotels. Good examples are the Smithsonian, Canada's grand railway chateaus, and Hearst Castle in California.
