I thought about that, but some of the results I found on Google suggest that output from WebGL shaders is supposed to be in a linear color space [0] [1]. Then again, some of the comments here [2] suggest that you do have to do gamma correction manually.
Incidentally, I didn't know to do gamma correction until I came across [3], which explained why the output from my raytracers always looked a little off ;)
If you load a texture like from a JPEG and use that in a shader, the texture is converted to gamma space when loaded, so you need no correction. If you are generating colors in the shader, you will need to correct for gamma.
GL does have the capability to handle linear color space buffers correctly, but you have to enable SRGB, and initialize the framebuffer correctly with SRGB color space, and I'm not sure if GLES (WebGL) can do this.
What happens if you need to alter the color of a texture that's been loaded? For example, if you want to do diffuse shading on a sphere with a texture mapped to it. Do you first need to convert the texture pixel back to linear space, apply the shading, and then correct for gamma?
I believe that, in WebGL, if you sample the texture in a shader the resulting color is in a linear space. And output colors from a fragment shader are meant to be linear as well.
I realize now that I can't fully answer your question, because I'm unsure whether WebGL implementations are supposed to respect color correction information in the image file format. But I'm pretty sure that the color space in WebGL "shader land" is always linear.
I have absolutely no experience with WebGL, so I have no idea. Sqrt would diminish intensity in highlights, if nothing else. So, in that way it would seem to be an approximation of gamma correction. I'd have to see the result (can't on tablet).
GL does have the capability to handle linear color space buffers correctly, but you have to enable SRGB, and initialize the framebuffer correctly with SRGB color space, and I'm not sure if GLES (WebGL) can do this.