Greasers are known for traveling in packs, cowboys are known for being alone, so one-on-one, I would have to go with the cowboy.
Cowboy: has to survive in the elements, in dangerous territory, with only his horse and a minimal amount of gear.
Greaser: a member of urban civilization, who presumably has a fixed shelter, does not have to know survival skills, and is dependent on modern technology.
Advantage: cowboy.
I have read recently (don’t have the book with me) that most cowboys never owned a gun- simply because they could not afford them (I am talking about 19th Century so you could suspend them. As for cleaner, they seldom bathed. All the Hollywood depictions of Cowboys are totally wrong- as are the numerous bank robberies.
Can we have a tag team match featuring cowboys and greasers, facing off against mods* and rockers?
for even better entertainment, make that Straight Dope mods. 
I’d parse the OP to be “ranch hand” vs. “street punk.”
The hand is going to be a hardbody who is familiar with blunt trauma and prolonged but steady exertion. He’s disciplined and will work through pain, and prefers not to quit before the job is done. His muscle memory is oriented toward himself–grappling, binding, lifting, balanced hurling. The punk is likely a sprinter with good agility but limited endurance and probably only average strength (what’s he do all day?) What he lacks in discipline and endurance he makes up somewhat in pride, so he’s not likely to give up unless he’s really getting his butt handed to him. His muscle memory is likely to be oriented more toward wielding bludgeons and blades, AND he’s got experience fighting people who know how to fight.
Cowboy/ranch hand is screwed. Punk comes out with fists, draws a couple punches to get the cowboy into a rhythm, the switchblade comes out stealthy and the cowboy starts losing elbow & knee ligaments. Cowboy is dead or crippled in under 5 minutes.
Why is it that your hypothetical ranch hand is unarmed? Actual cowboys have to contend with vicious wild animals and they carry guns. Let’s give the greaser a gun here too for the sake of leveling the playing field. How accurate do you think the greaser is with his gun? Assume we’re talking about handguns here. The greaser probably does not practice in any way with the handgun. It’s a weapon of last resort where if it actually winds up being used, in the urban environment, cops will be called and all kinds of heavy shit will go down, so he’s unlikely to actually have much experience with this gun. The cowboy, on the other hand, has a lot of practice because he actually has to use the thing on a regular basis, whether it’s to shoot wild animals like coyotes and wolves, or outlaws on the range or whatever. The cowboy wins.
I vote for whichever is the one who loves to get drunk and fight. Or plays hockey and loves to fight.
I knew a guy in Tokyo whose hobby was getting wasted then insulting US. Marines in bars. It never seemed to end well for him, but he loved fights, even outnumbered ones that would leave his face in TechnoColor.
I’ve only seen a few bar fights in real life, but most people don’t have a clue on what to do.
I think it’s in individual here, not the category.
He hasn’t got a gun because he’s a librul urban punk. And that’s totally germaine to why there’s about to be a fight with a [del]redneck[/del] cowboy. 
You were positing a situation where a “punk” (I don’t know how it got turned into “punk”, a greaser is a very specific type…it implies a certain degree of mechanical aptitude. It’s a notch above goons and punks respectively on the hierarchy of hoodlums) was armed with a switchblade but the cowboy was unarmed. I am merely contesting the plausibility of this scenario on the basis of a cowboy (busted down to “ranch hand” in your scenario) having familiarity with firearms as a necessity of the job.
I was looking for a synonym for Greaser that didn’t harbor Mexican baggage. On reflection I just switched it with some misfit armed with an orange mohawk. Not useful. Urban Ruffian, perhaps? Anyways, cowboys keep their knives in their boots. It’s a long way to reach and I can’t think of many ways he’d get at it without projecting his intentions. Cowboys have an honor code which, to me, suggests he wouldn’t open with knife if his opponent presented fisticuffs. Similarly, until faced with fire he’d leave his own on his hip. His opponent, on the other hand, has no sense of honor and will do what it takes to bring the altercation to a satisfying (for him) conclusion. Which means he’s going to be sneakydirty, probably to the point of coming out with at least a fist load if not a hidden zip gun.
I grew up in the country and then in a 'burb that had a 4th of July rodeo that went for close to a week. I don’t know how many times some cocky city kid tried to make fun of the cowboys/pick a fight and then proceeded to suffer a major ass kicking. (If unclear, it was the cowboys that kicked ass). Farm kids/cowboys generally are in much better shape, as pointed out above tend to have a very high pain thresh hold, and don’t give up.
That said, one of the bad asses at my high school would win his fights with the cowboys, and we were all careful not to get on his bad side.
Yeah, definitely has regional shading to it. Here a greaser is an extremely archaic term for a guy of any race who wore black leather, slicked back his hair (hence “greaser”) and threatened people he didn’t like with switchblades. I’ve literally never heard anyone use the term that wasn’t talking about The Outsiders or Westside Story. If someone did use it, I’d assume they’d also talk about soda jerks and poodle skirts and other stuff from the 50s.