Yes, there are cowboys in the US. Everything west of the grand Mississippi River in the early 1800's was unchartered and only the adventurous would dare leave the relative security of the colonies and those "territories" that, although not States, were up-and-coming with settlements and industry. So, the adventurous and those who had embraced the ideals of Manifest Destiny would cross the river in search of fortune, or at least a little plot of ground that they could call their own.
With little in the way of protections, the "settler" was faced with either retaliation from Indians whose land they were encroaching on, meaning "robbing the resources" that belonged to the Indian nation, or bands of brigands whos capitalized on the lack of protection and nominal numbers of any encampment.
Some of these encampments grew into towns, some of these settlers founded huge businesses of cattle and other livestock to provide food and raw material like wool and leather to the ever-growing United States. Those who had the affinity for vast open ranges and the wear-with-all to confront the open country became the cowboys of legend. They were far from the cowboys you would see on TV though. They were a rough a tumble bunch, and although some became very good with a gun, typically the Colt revolver, very few ever gained notariety for being gunslingers. Most were simply workers that lived and died on the range.
Today, there are still those revered men and women who work the range, ride their horses, and wield a rifle along with their napsack tied to the saddle. The rifle is probably not an old Winchester, more likely a Reimngton .30-06 with a scope capable of shooting a wolf or coyote at a few hundred yards. Little issues exist now between cowboys and Indians, most Indians have had their lands stolen by the Manifest Destiny playbook and have been confined to tribal lands called Reservations, which are little more than a concession given by the US for having pillaged their nation.
There are some good history books replete with accurate depictions of cowboys and Indians.....