It's less of an ethical reason than you'd think. We don't eat cats because they're carnivores. (Dogs are omnivores and I think that's why other countries do eat them.) Eating carnivores on a significant scale is a) economically difficult, and b) poses greater health risks.
First, economics, as shaped by trophic levels and food chains. As I'm sure you know, plants create energy through photosynthesis, which is ingested by primary consumers, i.e. herbivores. Carnivores can't eat plants, so they eat the animals which eat the plants, and so the energy produced by the plants passes up the chain. Simple food chain : sun's energy ->grass -> rabbit -> fox
For the sake of this post, we will imagine a variation on that food chain: sun's energy ->grass -> rabbit -> fox -> human.
The problem is that very little energy gets passed up at each stage. Approximately 90% of the energy (in the form of plantlife) ingested by a herbivore is lost to the food chain, so you need to give your rabbit 1000 calories of feed to get 100 calories of rabbit meat later. This isn't unique to the herbivores, so 90% of the energy (in the form of dead rabbit) ingested by the fox is also lost to the food chain.
This means that if us humans want to eat fox meat, we also have to give the fox 1000 calories of rabbit meat to get 100 calories of fox meat later, and now it's got expensive. How do you get 1000 calories of rabbit meat? You have to give the rabbits you're farming to feed the foxes, 10,000 calories of feed.
So you can use 10,000 calories of rabbit feed to produce 1000 calories of rabbit meat or you can throw 900 calories away to turn it into 100 calories of fox meat. Which will you choose in subsistence agriculture? It's less hassle and less money to farm the rabbits to eat ourselves, and forget about the foxes.
There are, however, some carnivores I can think of that humans do eat, because they farm themselves, so to speak: wild fish! This is where health comes in. Tuna are carnivorous, and they're also known for being high in mercury. Why are they high in mercury? Because they're carnivores towards the end of the food chain. This is called biomagnification, which means that a toxic substance, in this case mercury, accumulates in higher concentrations at every successive stage of a food chain. Algae absorbs mercury, which is eaten by plankton, which is eaten by smaller fish. The bigger fish get a dose of mercury from each of the little fish they eat. Then the tuna that eats that bigger fish ingests all the mercury banked in its flesh. Rinse and repeat, and the mercury content becomes high enough in the tuna that pregnant women are advised to limit how much they eat, and not to eat marlin at all. Guess what marlin eat? Tuna!
Most famously, this happened with the pesticide DDT and American birds of prey. DDT persisted in the environment, so non-lethal doses accumulated in the bodies of insects, which were eaten by insect-eating birds. The insect-eating birds were dosed with tiny doses of DDT with every insect they ate, which built up in their own bodies. Then that led to a population-crash in peregrine falcons, which eat smaller birds.
Here ends the info-dump!