During which war, lasted from1550–1600, was the first time that the Spanish in North America faced mounted Native warriors?

That would be the Chichimeca War.

https://en.wikipedia.org/wiki/Chichimeca_War

The conflict you're looking for is the Chichimeca War fought in northern Mexico (part of Spanish North America). Horses were originally brought by the Spanish, but some escaped or were captured. Indigenous groups like the Chichimeca quickly adapted and began using them in warfare - turning Spanish cavalry from an advantage into something much less decisive. The war frustrated the Spanish a lot more than they expected - it dragged on for decades and forced them to switch from pure military conquest to negotiation and incentives.

During the war, the Chichimecas learned to ride horses and use them in war. This was perhaps the first time that the Spanish in North America faced mounted Native warriors. The undeniable advantage for the Spanish was their use of horses and other animals of burden that they had introduced to the Americas. Horses were extinct in the Americas before the Spanish reintroduced them in 1519.

Source: https://en.wikipedia.org/wiki/Chichimeca_War