While Virgil Earp was, indeed, a veteran of the Union army, Neither Wyatt, nor his younger brother Morgan were (both were too young at the time). This actually worked in Wyatt's favor, when he became "peacekeeper" in both Witchita, and Dodge City, during the cattle boom.
Since the majority of the herds arriving for sale in these boom towns came from Texas, and were driven by ex-Confederates. The "powers that were" *wanted* these ex-Confederates to hang around long enough to put most of the herd sale money back into their pockets, in the form of entertainment for these cowboys who had spent long, hard months driving the herds to these market towns. A local lawman with a Union Army background, with an obvious axe to grind, didn't fit in with these plans.
Enter young Wyatt Earp, an ambitious, rather mercenary, young man, without *any* of that baggage. He became a good fit as their "town super-bouncer", and I'll admit, he was quite successful at it, moving from one boom town to the next. But in the final analysis, his actual "career" as a lawman reads more like the career of a Patrick Swayze "Roadhouse" style bouncer than anything else...
the infowarrior
You're right of course, Virgil was the only actual Union veteran of the Earp brothers. But I think Virgil's Union veteran status had at least something to do with the enmity between the Earp clan and the Clanton/McLaurey bunch. OTOH I don't think that either group's hostility toward the other was primarily motivated by loyalty to one side or the other of that fratricidal war.
I tend to believe that both cliques were motivated more by monetary greed and a desire for raw political power than by any other factor. Neither faction was worthy of the adulation heaped on the Earp's and Holliday by the eastern press and the dime novel authors.