We are simply too smart and too well-armed for any wild animal to hunt. Now consider the poor zombie. It lacks every single advantage that has kept humanity from being eaten to extinction. It wanders around in the open, it can't use weapons, it can't think or use strategy. It doesn't even have the sense of self preservation to run and hide when it's in danger. And, it's made entirely out of food. It's easy prey for any animal that wants it.
If you're saying, "Sure, but it's not like my city is full of bears that can come eat all the zombies," you need to think smaller. Insects are a major pain in the ass for living humans, and in some cases, being able to swat away flies and having an immune system is the only thing keeping us from having our eyes and tongues eaten out by maggots. Zombies in any part of the world with a fly problem are going to be swarming with maggots in short order, meaning that most of their soft tissues will be infested, and their eyes will be very quickly useless.
Not so disgusting now, are they? OK, yeah, but show a little respect.
We'll scale up a bit: In America alone, we have bears, wolves, coyotes and cougars, all of which can put well-armed, thinking, fast-moving humans on the menu, if the conditions are right. To most predators, the "right conditions" are when the animal is weak or infirm, or otherwise generally unable to defend themselves, like a walking corpse. Hell, just think of the millions of stray dogs out there who'll quickly learn that zombies are an easy meal.