Government and Warfare.

Here's another, more specific checklist distilling what we know for sure. I do a little more theorizing for the pages that haven't been covered, but I'm mostly trying to speculate using historical precedents. For the most part, the type of troop that dominate roughly correspond to how they're governed--monarchies and warlords use knights, republics use pikemen, although you could also classify it by terrain--mountains and cities for pikemen, open woodland for knights, and plains and deserts for horsemen.

Secretarial States: White's own invention, found in the Caribbean island and adjacent continental territories, where women run the civil government and households.

Hydraulic Empires: Not necessarily invented by White, but a relatively uncommon term who's definition is still being sussed out. They are absolute monarchies enabled by a top down control of water, usually found in arid regions.

Tribalism: This was probably the default for North America thousands of years before the arrival of European settlers, and in Medieval America favored by denizens of the continent's center, due to their nomadic lifestyle. The American are strictly pastoral, but some Hunter/Gatherer types look to exist in Canada (and likely Alaska).

Feudalism: Feudalism is almost synonymous with "Medieval", particularly Europe, and commonly used to describe pre-modernized Japan. It's somewhat new to America, although people have invoked the word to describe the Patroon system of New Amsterdam, the seigneurial system of Quebec, and the Antebellum South. In fact, White mentions contemporary plantations and we may have seen something splits the difference between them and the old school medieval manor.

Republic: The United States was itself a republic, of course, and the vestigial U.S. remains one, at least nominally. They are also found in the independent city states of the Pacific Northwest, as well as New England. Medieval precedents would include the Italian city states and the Swiss Cantons (Which bore some resemblance to Colonial New England's town halls). It's possible White would have gone into why some parts of America remained republican, and it would possibly be chalked up to things like continuity, urbanization, an economy based more on finished goods than raw agriculture, and the relative difficulty in establishing a land-based aristocracy.

Unlike Government and Religion, it appears none of the Military units have been invented, or even creatively subverted by White.

Horse Archers: The only one of the four pages for Warfare that White created, it's actually the style of military combat most familiar to Americans as it was the medium of choice for Plains Indians, even well into the industrial revolution. 

Lancers: Or knights, or heavy cavalry, as emblematic of the Middle Ages as they are alien to America. From White's illustration, the knights seem to resemble those of Easter Europe and Asia minor more than Western European knights, though the profile also hints at something akin to the conquistadors and armored soldiers of the Colonial era. White makes a reference to sabers, which may be the primary type of blade--considering it was the sword most often used in America's history, (though mostly for ceremonial purposes) as well as the sword of choice for eastern Europeans--who's climate is most similar to the American Midwest. Knights largely dominate feudal states, but can also be found in the hydraulic empires (Which, considering the climate and ostensible celibacies, evokes Holy orders and Crusader states.) They're also the dominant in the core U.S.A. territory, and it's interesting to ponder the difference with the warlords.

Pikemen: Pikes, which are usually distinguishable from spears and other polearm type weapons, were the favorite medieval weapon of armies from places like the Italian City States, the Swiss Alps, and the Scottish Highlands--all areas with Medieval American counterparts like New England, the Appalachian Mountains, and the Pacific Northwest. In the "trying to resemble early America as much as the Middle Ages" trend, it's possible White's version of Pikes might be similar to bayonets. Pikemen also seem to be the main force in New Jersey, which a feudal state, but part of the urbanized, pike-using Northeast.

Navy: It seems that the United States of America (which as a political unit, is more a successor to the U.S. Navy than anything) is the supreme naval power of Medieval America, and has all but a monopoly on the east coast. Charleston, South Carolina seems to be the only major east coast city not under its control. (Though that may also include Boston) The Great Lakes would also see more military action than its previously peaceful borders are used to. While Naval battles are not new to the continent, the Medieval way of boat-ramming would be a little more unknown territory. and to fill the void left by canons or missiles might tempt them be cockier with fiery arrows.