If your picture requires a magical display to "work," if it requires super resolution or a color calibrated monitor or it only "works" as a print, or only as a Fine Print then it doesn't work at all.
Sure, you can ruin any picture with enough abuse, but the functioning of a picture is in the stuff that survives mild abuses just fine.
Show me a great picture, in any medium, that isn't still pretty great 1000 pixels on the long side on the web. Go ahead. Think me up one or two of the iconic paintings and photos that just turn to crap on the web.
If your pictures look like shit on the web, it's because your pictures look like shit, not because of the web. Stop making excuses and go shoot some pictures that don't look like shit.