Oh my, no. What you probably perceive as the traditional American "Cowboy" has its primary roots in (first) Spanish culture, and later in South American culture via the Gauchos of Argentina and Uruguay, but most especially, the Vaquero of Spanish Mexico (mid-to-late 18th century). u/DBHT14 touched on this quite correctly. Many of the wardrobe and "rig" accouterments of the classic American cowboy originated from and was heavily inspired by the clothing and gear the Spanish Vaqueros sported.

If you are looking for information or more history about what gave rise to the almost mythic American "Cowboy" archetype we are familiar with today, I can suggest a couple of great books. You might start with Daryl Jones' 'The Dime Novel Western,' which traces the history of popular representations of the classic Cowboy figure throughout the mid-to-late 19th century by examining Dime Novel literature. Jones ties the rise of the classic Cowboy in popular entertainment (first Dime Novels, then radio and film) to social developments in the U.S. during this period (which include the Great Railroad Strike of 1877, the Haymaker Riot of 1886, etc.). He also talks about the establishment of the American West as a mythic land of plenty, and how this image was framed in such a way as to require civilization. To have something that needs civilizing, you need a threat (non-white people, scary beasts, criminals), and mythic storytelling tropes therefore require a hero: one unique to the setting (hence the Cowboy).

You might also check out Michael Denning's work, 'Mechanic Accents,' which ties in a legal discussion about the Indiana State Supreme Court ruling regarding "No duty to retreat" as a heavy influence on the socio-political climate of the region and era. This gave rise to a unique legal condition in many parts of "the West" that allowed people to justify homicide on entirely legal grounds (setting the stage for the classic Cowboy violence trope so important to the archetype).

I only bring this literature up to frame my answer a bit better, because to understand the origins of what is popularly regarded as the classic American Cowboy, it helps to understand the social, political, racial, and even legal elements involved in its formation from the roots of Vaquero and Gaucho origins.

EDIT - If anyone would like more information on this subject, please feel free to leave a comment or message me. This particular field is what I got my Masters in, so it's nice to be able to chat about it (I mostly field Civil War questions here, so it's a nice change of pace = )

Follow-up question: Were there any "Cowboy cultures" outside of the Americas, or was it unique to those two continents?