native American force

Neither the Spanish nor other European powers ever sent an army to the Americas in the initial stages of conquest.

Mongol horse archer army

No.

Did traditional pike and shot armies ever fight all infantry or all cavalry armies in the open field?

The closest was probably during the Hussite wars (1460s-1460s), although it was a bit too early for the early-modern pike and shot. The Hussites had a mostly peasant infantry, but they still had pikes, firearms. They had little to no cavalry. They faced a composite crusaders' army with heavy calvalry and also other arms.

To overcome their deficiency in cavalry, the Hussites developed a circle-the-wagons strategy to be able to fight in relatively open fields.

By this point, it was already known that combined arms are a powerful concept and nobody willingly fought in handicap fashion.

In the case of the New World conquest, although some members of the expeditions had certainly fought in the Italian Wars (1494-1504) where pike/shot/word formations were utilized by Gonzalo Fernandez de Cordoba against the French, the expeditionary forces did not really fight in a pike and shot formation. It appears from Diaz del Castillo's account of the conquest of Mexico that most of the soldiers were armed with swords and shields. At that point in the early 1520s swordsmen were still a part of Spanish formations, which would become known as the tercio. And of course, pikes don't really work when you only have a few hundred men and are frequently surrounded by enemies, so having what I will call a "sword and shot" formation was more useful.

As for cavalry vs. pike and shot, I don't know of any wholly cavalry armies that attacked pike and shot formations, but mounted elements of an army certainly have, without being supported by infantry or artillery. In short, these cavalry charges have not gone well, cavalry after the reemergence of pike formations in the 15th century cannot generally take on infantry from the front, and need to be used against demoralized troops of from the flanks. (all of these examples happen to do with the French because I like French failures as a Spanish historian)

Battle of Cerinola (1503)- French cavalry under the Duke of Nemours charge towards a thin line of fortified arquebusiers backed up by pike formations. The charge is stopped by shot and the ditches, and then the pikemen advance and push the cavalry off the field. Not an open field but it was a field battle with hastily erected defenses, which was one of Fernandez de Cordoba's trademarks.

Battle of Pavia (1525)- King Francis charges ahead of his infantry support with his heavy cavalry, but is essentially ambushed by pike and shot elements coming out of the woods from all directions, he is captured and many of his knights are killed.

Battle of Waterloo (1815)- Pike and shot has been combined by giving everyone bayonets- Marshal Ney charges with 9,000 cavalry at Wellington's cannons and infantry. They overrun the cannons but the Brits fall into battalion sized square formations and repel repeated assaults.

It wouldn't have been an all cavalry army but the Swedes faced off against very cavalry heavy armies during their wars in Poland in the 1600s both prior to the Thirty Years War and during the Deluge period.

There were battles in the Highlands in Scotland during the English Civil Wars when the government forces were going after Montrose that were largely infantry battles with very little cavalry on either side.

Although it wasn't exactly an army studying King Philip's War can give you an idea of how soldiers with equipment of the pike and shot era dealt with Native American enemies.