When the U.S. was expanding they took over part of Mexico (and fought a war to settle or border where it is now). Considering that it was considered "ok" morally back then, why didn't the U.S. concur Cuba and the Caribbean islands, eventually making them new states like they did with Texas?
By today's standards that would probably be the most valuable land in the country.
I won't touch on Cuba since I don't know a whole a lot of that particular island. But the US did expand into some Caribbean islands (Puerto Rico which was taken from Spain, and the Virgin Islands which were purchased from Denmark in 1917, Navassa island) but the vast majority of the Caribbean was owned by the Netherlands, France, and Britain. All nations who the United States didn't want to go to war with.
Edit: Forgot to mention filibusters, but someone else did.
By the middle of the war against Mexico domestic opposition was firmly against further territorial expansion into any territories that had the potential for future slave states. This opposition first emerged when the Republic of Yucatan appealed for annexation by the United States quickly erupting into a major political brawl that even Democrats were loathe to touch. There were American filibusters(Americans who desired to expand America by extralegal force of arms) who attempted to expand into the Caribbean and Central America but they were opposed by the Federal government and despite what you may hear on reddit actual Southern support for filibustering was very limited in the time period you speak to. I have spoken at some length regarding territorial expansion in Mexico and Yucatan in previous comments and I'll try to find the links to them when I get home from work.
(and fought a war to settle or border where it is now).
Not to nitpick, but this isn't quite the case. While the Mexican-American War added California and most of New Mexico to the United States at their present borders, the 1854 Gadsden Purchase set the Arizona-Mexico border where it is today.