The vast majority of the 16,000 U. S. regulars were stationed in the west prior to the Civil War. They had a lot to do: keep the mail coming, keep the Native Americans from killing folks, and keeping the peace in the new state of California, which was geographically HUGE compared to the states in the east. It still is, btw.
The regulars went east as soon as Fort Sumter was fired on, so the new recruits took their place in California and the territories of New Mexico and Arizona.
On April 15, a very small group of Union cavalry (12 men) attacked an even smaller group of Confederate cavalry (10 men). This kerfluffle took place about two miles from Arizona's Picacho Peak, and the skirmish is considered to be the western version of the Confederacy's "high water mark."
Unlike Gettysburg, the Union lost this one. Just like Gettysburg, men died.