Yes but there would European style of armies in each section of North and South America, this setting has to deal with 19th Century American Westerns setting.
LW
Define "European style". There were plenty of European style armies in the Americas OTL--the Chileans were trained by Prussians, for instance, and both the USA and CSA in the Civil War was deeply inspired by Napoleonic warfare. But for the "19th century American Western" setting, those sorts of armies are difficult to work in the wide-open endless plains where mobility is king, so the organisation would be closer to the historic US Army in the West, although a bit bigger perhaps and with more infantry/artillery support not needed as much OTL in the Indian Wars. I could imagine a campaign on the Great Plains in a major war would be effectively like Operation Barbarossa, and the earlier you get in history, the harder it's gonna be--they didn't call it the "Great American Desert" for nothing.
The Rockies and Great Basin are a bit different, of course, but the same principle applies. Actually, for the defender, it's even better, since the settlements are very sparse and spread out, with ample terrain for ambushes on supply lines and such. For California, it's easier because the land is better, but California still has tons of rough spots and
many chokepoints (all those mountain passes) for easy ambushes and strategic battle sites. Trench warfare (either the late Civil War style or WWI) would be highly effective. You also would have to get supplies
to California, so a naval war is essential for resupply unless you want to send it from Northern Mexico across very harsh deserts. The Apache live in that region as well, I might mention. The Yaqui people also live there, and they were among the last native groups subdued OTL.
So pick your firearms accordingly. Preferably you want light and mobile forces, and probably mountain infantry as well. California is the only exception.