The short answer is, it's not right.
The longer answer begins: What do you mean by organized crime?
Criminologically, organized crime is very similar to Justice Potter Stewart's famous quip about pornography--that it was difficult to define, but that he knew it when he saw it. With organized crime, one commission defines it one way, another academic defines it some other way, and still another agency has a different definition. Letizia Paoli's paper, 'The Paradoxes of Organized Crime," was written about 20 years ago, and talks about this at length. It's available for free, and I urge everyone who 's interested in organized crime to look at it: Paradoxes of Organized Crime.
In any event, we all seem to agree that it involves a group of people organized on a long-term basis in order to make money in some way that violates the law. Other elements are added to that definition based on who's talking about whom and what.
So who were the Cowboys?
The Cochise County Cowboys were a group of cattle and horse thieves that coalesced in the late 1870s and operated along the US-Mexico border into the 1880s. (Some people know this and some people don't, but if you don't-- yes, Johnny Ringo and Bill Brocious were real people, and the gunfight at the OK Corral was in 1881) The border was crucial to what they did. The would steal a group of cattle or horses in one country, take them across the border (where law enforcement couldn't pursue them), and sell them in the other country. That was their bread and butter, and that's why they needed to be organized. To steal horses and cattle at the quantity in which they stole them--sometimes hundreds of head at a time--a lot of people are required, and they all need to know what to do to make the operation work. You're going to want to have people who can tip you off when people are driving their herds in places that are convenient to theft, you need to have places to sell them, and it'll certainly help if there are some individuals within law enforcement who are willing to look the other way when you need them to. This is in addition to all the ropin', ridin', and rustlin' kind of skills that a person's going to need in order to pull off the actual theft. Almost no one would dispute the idea that the Cowboys were an organized criminal gang.
But nearly contemporaneous with the Cowboys was the Black Hand, a group (or style) of Italian kidnappers and extortionists that are considered to be the forerunners of the Mafia and operated within Sicilian and Southern Italian ethnic enclaves throughout urban America during the late 19th century. A lot of people will firmly peg the Black Hand in the US to the 1880s, but similar organizations existed in Southern Italy going all the way back to the mid-1700s, so I find that less than credible. Regardless of when the English language media in the US took notice of the Black Hand, they were most likely concurrent with the first Italian neighborhoods in the US.
Regardless of whether the Cowboys or the Black Hand came first, there were certainly gangs in the US that predated both. The Live Oaks were an Antebellum gang in New Orleans that used to hang out at the bottom of the French Quarter, who rolled and pickpocketed boatmen coming in from off the docks. They engaged in revenue sharing and division of labor, and persisted regardless of the incarceration of any one individual. Of course, there are famously the gangs of prewar New York like the Dead Rabbits, chronicled by Herbert Asbury in his seminal (and highly dated) The Gangs of New York: An Informal History of the Underworld. Asbury also did books on the old-timey underworlds of New Orleans, Chicago, and San Francisco. If you're interested in this topic, they're all good books. (At least in the sense that you can find obscure information described colorfully and in detail. None of them are academic; all are offensive to modern sensibilities. The drollness with which Asbury talks about things like child prostitution is quite disgusting to modern ears, but these books were written during the 30s and 40s, when people didn't even go to jail for lynching, and raping your spouse was legal. It is what it is.)
I don't see much of a distinction between the gangs of the 1800s and the organized crime groups of the 1900s or the present day. The main difference is less one in the behavior of the outlaws and more a function of the law. In 1800, almost nothing--emphasis on *thing--*was illegal. Prostitution, slavery, drugs, booze, copyright infringement--all legal activities in 1800. Even actions like rape, theft, or murder were legal if the recipient of the violence was nonwhite. If a person had the urge to make a living by inflicting harm on others, all they had to do was go to the frontier, or save up enough money to buy a slave. Where's the black market to take advantage of?
I would argue that the first large scale illegal market in the US is really slaves from Africa. When the importation of slaves was banned in 1808, it's the first time in the history of the US that a whole market--in this case, international slave trading--had been driven underground. This date also coincides with the Embargo Act, which prohibited trade from France. These two events helped to provide the setting for success for Jean Lafitte.
Lafitte, who is both celebrated and euphemistically referred to as a "pirate" is probably the first high-profile person in US history who is basically indistinguishable from a modern-day mobster. In reality, his bread and butter was people smuggling just as much as the Cowboys' bread and butter was cattle smuggling. He ran numerous ships, had a whole network of people inland to smuggle the cargo into the US, exercised a lot of pseudo-political influence, and so-on.