Would they eat wild horses? Yes, and they do. Would they control their populations? No, not really.
The fact that there were 30 - 60 million bison roaming North America before westerners came to the continent should be an indicator of how well predators here are of "controlling" populations. Granted, wolves and bears probably preyed more on bison than cougars did, though there are early accounts of cougars riding the backs of bison.
The summer diet of large male cougars consists mainly of large mammals, including moose, deer, and where available: feral horses.
The thing with cougars is that they are solitary predators, they don't live in prides like African Lions, and they have huge territories they defend from other cougars, so comparatively, they aren't killing that many horses.
Cougars are also hunted themselves, and their populations are controlled, because like any other predator: they like easy prey. Given the opportunity to choose between a feral horse running wild, or an old captive horse stuck in a fenced field, chances are they'd go for the horse that couldn't run away.
There is a balance in nature between predator and prey. They amount of available prey determines the population of predators it can support. If there are too many predators, then the food source dwindles to the point where the predators starve, hence why many predators are territorial, they defend a territory sufficiently large enough to sustain themselves, and defend it from other predators.
If there were a sufficiently large enough population of cougars to control feral horse populations, then there would be a lot more conflict between ranchers and cougars as well, which in turn would result in the culling of cougar populations.
So the answer is "Yes" cougars can and do take down feral horses, but they won't take down enough to solve the problem ranchers have with feral horses; That problem being that feral horses destroy fields of crops, damage fences, and eat the feed intended for the herds of domesticated horses and cattle.
What will happen with the horses, if left on their own, is that a balance will be achieved. More horses means more potential food sources for cougars and other predators, but it also means more competition for the other animals that eat the same food as the horses. The horse population will grow as much as the availability of food and predation will allow. The problem with the horses is that they are an invasive species. Native ungulates may have difficulty competing with horses, which would result in a decline of native species. The cougars and other predators may be able to help balance the overall population of prey, but they wouldn't target the horses specifically unless they were an easier prey.