Can you drown a fish?

Well if we use one of Mr. Webster 1913's definitions:
"To deprive of life by immersion in water or other liquid."
ba's idea (which has been nuked, but went something along the lines of jamming a hose down the fish's mouth and pumping it full of water until it explodes) makes sense.

However, the more standard definition:
"To be suffocated in water or other fluid; to perish in water."
applies to fish, too.

Fish, like most forms of life, need oxygen to survive. As you may remember from biology classes, fish extract oxygen from the water as they swim by passing it over their gills. Anyone who has kept an aquarium knows that too many fish in a limited volume of water will eventually extract all the oxygen from their environment. They will begin gulping at the surface (some fish like bettas and gouramis have actually evolved the capacity to survive in this manner), and then suffocate due to lack of oxygen.

Thus, the poor fish drowns.

I would add that since, as wunderhorn1 points out above, fish extract water by passing it over their gills, they can be suffocated (in many cases) by preventing them from moving in relatively still water. Sharks are the most extreme example; their oxygen demands are too high to permit them to circulate water with their gill opening muscles as many fish can. For this reason, unlike most fish, sharks cannot stop moving. If they do, they drown.

This also means they cannot sleep, and in fact do not. This goes a long way to explain their legendary irritability and generally lethal attitude.

I didn't want to add a writeup here (in fact, I sent a /msg to ba about this, as soon as I read his), but:

You can drown a fish in oil. Olive oil, corn oil, 10W30, what have you. In fact, you can drown a fish in any oxygen-poor liquid, including water itself!

But please, don't try this at home, it's cruelty to animals!

There is, of course, a more figurative meaning to this: "Can you drown a fish ... in tartar sauce?" YES! You might really like tartar sauce a lot, but don't overdo it! If you're a fan of salt or hot sauce, take care not to drown your fish (or any other food, for that matter) in those condiments either.

On the idea of drowning fish...
My mom, the grammarian, often found herself sputtering when she'd become angry enough. She'd yell at us kids, confuse our names, spout enraged nonsense statements composed mainly of four syllable words, arcane phrases or Catholic threats of damnation.
My brother was one of those kids who have a zillion cages of critters and tanks of fish. He was a little lazy, however, and on occasion would be less than fastidious about cleaning the tanks. I heard her in his room one day, hollering at him to clean the fish tanks. As she stepped from his room, she turned back to deliver one last threat.
"If you don't keep those tanks clean, I'm going to drown all your fish!"
Both of us knew better than to laugh.

Betta splendens, commonly known as Saimese fighting fish, rely heavily on their labyrinth organs for oxygen and as such can easily be drowned in oxygen-poor water if they are prevented from surfacing to gulp air. I once had a betta who got trapped under a plant in his tank and subsequently drowned as he was unable to free himself and reach the surface. Even though there was a bubble stone in his tank, he was unable to get enough oxygen through the sole use of his gills.