Hang on - as has been pointed out, they definitely did make grabs for Cuba and Mexico. The problem was that most of the previous American expansions - Texas and Louisiana being the textbook examples - meant expanding American sovereignty over people who wanted it (at least if we ignore those of them who weren't white or, well, not Mormons), or could at least comfortably be imagined becoming part of a (white, basically Protestant) American nation.
That was my point: the US could have moved to annex Mexico and Cuba after its war victories. They
refused to do so because they didn't want the mexicans and cubans that would come with it. Even when those wars were launched the intention was to create satellite states. In the case of Mexico also consolidate territorial grabs of what was at the time marginal lands. Seriously they only tried (and succeeded) to annex the west (they managed that), and Canada (failed, the British Empire at the time was strong enough to give them pause).
I'm not sure what you're saying about monarchies. Are you saying that they were less prone to land-grabs? If so, the British might want a word, given that the Oregon dispute (in which the US came close to demanding most of the populated bit of Canada) concluded just about a decade before Britain formally annexed India, and acquired two and a half million new subjects. If you're saying that they were more successful about it, the British might again complain about their ongoing frustrations in Central Asia or in Africa, where they were hardly able to expand unchecked.
No, I meant that monarchies were more prone to land grabs. Even the obviously dangerous ones: the Hapsburgs annexing into Bosnia, or the russian tsars grabbing all of central Asia. The idea was that "national loyalties" did not matter, sovereigns just grabbed or swapped around territories and populations, whether by war or marriage. And that had been the case. Until "nationalism", we may say, changed that. But it was not really nationalism but one of its components, arguably the most important one: "popular sovereignty". The population were no longer passive subjects of a monarch. They were now the main actors of the political unit. And the political institution for these states was the republic.
The US, being a republic, had to recognize popular sovereignty: sooner or later its territories would be organized into states and the representatives of its populations would gain seats at the high table. Problem was, those already sitting at the table were horrified at the idea of having
mexicans sitting on it! Filthy catholics, peasants, layabouts, and whatnot... worse, they'd not be just a few, but rather a lot, enough to become a major power block. Thus mexico could not be annexed, the ruling elites would rather refuse to take that territory.
This has happened in several other instances, even in the breakup of the european empires. Only the french kept a few possessions giving the local population the vote, but only because those were small enough that they'd not sway politics in Paris. Keeping
Algerians as french citizens and letting them vote parliamentarians was equally horrifying to any frenchman, which is why De Gaulle terminated most of the french empire. Ruling elites don't want to share their power with unpredictable newcomers from assimilated territories. So, only territories with sparse populations (and few representatives to appoint) get annexed. The others may be kept occupied as long as feasible (diplomatically, militarily) but are not formally annexed. The obvious present example of this is Israel.
Also, as TF has pointed out, the early American Republic was considered essentially a monarchy - certainly not a democracy. The executive powers of the President are essentially identical to those of the British crown, and there was a colossal fuss over what to call him. John Adams pressed for 'Your Highness' or 'Your Majesty': the alternative was 'your Excellency', but Washington eventually plumped for 'Mr President' not because those were too grand, but because a good number of people thought 'Your Excellency' wasn't grand enough!
It was different from a monarchy in the very important aspect that the population of its territories got to elect representatives to sit at the table where power was wielded: it was part of american political ideology and could not easily be put aside. Monarchies like the British Empire (and other european states at the time) did not allow that to its imperial provinces. I think the British Empire only ever tried it with Ireland, and failed. Other bits it would rather cut off (turning developed colonies into Domains, quitting colonies) rather that grant them proportional representation in London.