I am aware of one US Army experiment involving the importation of camels for field work that showed promising results before being cut short by the Civil War, but that seems a little too late in comparison to the ubiquity of Horses in the Americas now. Considering how Camels are well-suited for desert environments and the prevalence of deserts and dry areas (Great Basin, Patagonian and Atacama deserts, etc.), wouldn't it have made sense to bring them over as well? Also, wouldn't the camel's ability to survive extended periods without fresh water make them just as easy, if not easier, to ship across the Atlantic than Horses?
Great question, but it is based on some of the same misconceptions that inspired bringing camels to the American West in the first place: namely, that just because people use the term "desert" to describe areas of the American West, they are actually different from most of the desert habitat of camels in the Old World. That said, there were experiments, and they were of limited success while generally representing failures. The following is a text I wrote on the subject for the Online Nevada Encyclopedia:
In 1856, the U.S. Army experimented with the importation of Middle-Eastern Dromedary (one hump) camels. Lt. Edward F. Beale maintained the animals could haul materials for the military in the arid West. They could carry more than horse or mules, and they had a legendary ability to survive without much water. Beale and roughly thirty camels crossed the Colorado River into present-day southern Nevada in 1857, but his experiment eventually failed, and the army sold what stock survived.
There were other attempts at importing camels including the 1861 introduction into Nevada of nine Asian Bactrian (two-humped) camels. Three years later, Sam McLeneghan purchased ten of the Army's Dromedaries for hauling supplies in the territory. Camels brought salt to mills in both Virginia City and Austin.
The animals generally proved effective, but operators had to avoid paths with sharp rocks, and they found that camels frightened horses. In 1875, the Nevada legislature prohibited camels on public highways to safeguard horse traffic. This effectively ended the commercial use of camels.
Operators set many camels free while selling others to circuses. For decades, various people throughout the West reported seeing the wandering beasts throughout Nevada and the southwest. Although some of the initial accounts may have been based on reality, these stories eventually became part of western folklore. Camels later assumed a different role in Nevada history and culture. In 1959, the revived Territorial Enterprise reported the results of a fictional camel race held in Virginia City. To the delight of residents, the San Francisco Chronicle reported the event as fact. The following year, actor-director John Houston, in northern Nevada for the filming of The Misfits, heard of the contest and became determined to ride in the "second annual" camel race. Virginia City held an actual competition, Houston won, and the annual camel races grew into a tradition celebrated to this day.