If you're neutrally buoyant, there is not gravity.

Yes there is... Gravity is gravity is gravity. If I jump in the air does that mean there's no gravity while I'm going up?

This is the fundamental difference between a fish floating in water and a person jumping up. http://i.imgur.com/nkwVc.png

In the second case, gravity acts on you even when you are going up, in fact it is what slows you down and then makes you start to fall.

Yes they aren't identical situations, but you'll notice in your diagram that there is a downward force acting on the fish called "gravity", which is the point I was making - that gravity is acting on fish and humans regardless of what upward forces are also present.

Try to appreciate the sense in which his statement might be true before nitpicking. It's true that free-fall feels substantially different from floating underwater, or even lying on a soft surface (the last two are more similar subjectively than free fall - the force resisting gravity is spread over your body).

But the swim bladder which makes them neutrally buoyant will give them an "up" and "down" sensation.