Given the hugely different perspectives on what the government of America should look like, and the fact that a republic of that type had never existed, it's entirely possible that America could have become some sort of monarchical government had things gone a different way. While it sounds outlandish nowadays, everything was far more uncertain. There were groups calling for Presidents serving life terms, many wanted George Washington to serve a third term (and Washington won both of his elections by unanimous vote), and the limits to the power of the federal government of the United States were not decided. The fact that John Adams peacefully handed control over to Thomas Jefferson when he lost the election of 1800 rather than using the military to seize control was considered surprising by many. So, it's not entirely outside the realms of possibility that, had things gone a different way, America could have become a constitutional monarchy. Of course there would have been opposition, but the main opposition to Britain was that they were being governed without representation.
The monarchy thing would lead to much stronger and more centralized federal government at the expense of the relative autonomy and power of state governments prior to the American Civil War. There would be opposition to this, definitely. But American belief in ideals of self-governance, liberty and representation means that the monarch, if there were one, would probably be little more than a figurehead, a Head of State to cooperate with the Prime Minister as a Head of Government - separation of powers. Or it would manifest itself in something similar to the relationship between the President and Vice President today. There would be checks and balances, and there would still be two houses of government, the House of Representatives elected popularly, and the Senate appointed by government (as it was up until the early 20th Century in the United States). The Prime Minister would either be the head of the party (as it is in most Commonwealth countries, e.g. Britain, Australia) or a nominee of the party (as is the case in America now).
So on the whole it's entirely possible that the USA could have become a constitutional monarchy (although it may seem unlikely now that it could have gone that way), at least I think so.