The challenge, with a POD no earlier than 1520, is to have independent monarchies based in the New World ruling over a majority of its population, land area, and total number of states. Personal unions with Old World monarchies are out, unless the New World nation in question is the dominant partner. Constitutional monarchs are acceptable.
OK, let's grab the easy ones: Brazil and Mexico. Somehow, both Brazil and Mexico retain their monarchies to the present day.
There are a few other options for establishing monarchies in the New World. An alternate Inca conquest could leave an Inca emperor as a vassal of Spain (probably with his title translated to something lesser), then become a figurehead for an Inca revolt. The Aranda plan - Spain establishing lesser princes as kings in the New World - could lead to those nations declaring their formal independence later on.
That just might get most of Central and South America under monarchic rule, which could be enough to fulfill my above criteria. But let's go further. What would it take to get monarchies dominant in *North* America?
I doubt France or Britain (or the Netherlands, for that matter) would be inclined to establish monarchic rule in the New World. One option is the classic "king escapes to the colonies after a revolution" trope. Another one, perhaps more stable, is a disputed succession, where one claimant gains the support of the colonies, fails to win at home, and sets himself up in the colonies, eventually leading to a treaty that formalizes the situation. Think the War of the British Succession from LTTW, except if Frederick was unable to eliminate William IV, but William couldn't take the colonies. Finally, we could get a powerful governor setting themselves up as a king, but that's harder to make stick the later the rebellion happens.