Various meats can be "cooked" using lemon juice. But how does lemon juice actually "cook" the meat?
Let's start with a definition of cooking.
The Oxford English Dictionary defines to cook as:
However, from a chemical point of view, what happens when you are cooking meat is that you are using heat, amongst other things, to denature the proteins of the meat. Denaturation can also be achieved by other means, such as marinating the meat in an acid, such as citric acid (contain in lemon juice) or acetic acid (contained in vinegar).
You can also use other agents to induce denaturation, but they are not always fit for eating.
However, it is important to understand that from a purely cooking perspective the two things are not the same: "cooking" with lemon (I guess it is fine to use the verb to cook as long as you put it in quotes), gives a completely different result than cooking with heat. This is because other chemical reactions happen during application of heat and because, obviously, you do not end up with an hot product!
During cooking, heat, amongst causing other things like the Maillard Reaction, denatures (changes the structure of) proteins in meat. Acid is also a denaturant, and so affects the proteins in the same way.
However, a mild acid like lemon juice isn't strong enough to kill bacteria, and it of course only affects the parts of the meat it can reach: consider something like beef carpaccio - when you slice it, you can see that only the surface layer has been denatured. There is also no Maillard Reaction, which is extremely important for both flavour and colour.
Cooking is only through the application of heat.
Since stomach acids can't kill 100% of pathogens, who is crazy enough to believe lime or lemon juice can?