Untested across the whole gamut of browsers, but if you're using the OOTB DOM, you have to do much more than just set the quick launch height. The
#s4-leftpanel-content div contains the content and background color of the quick launch, so this has to be given a height of 100% (which sizes it to the height of the
#s4-leftpanel needs to be given a height of 100% to size it to the height of
#s4-mainarea, and now you've hit the roadblock.
#s4-mainarea does not fill the screen, it sizes to fit the content. Furthermore, you can't just give this element a height of 100% because it's floated inside
#s4-bodyContainer and contained within
You could set it dynamically on load with jQuery (grab height of
#s4-workspace and set the min-height of
s4-leftpanel-content to that value), but that often leaves the awkward pause between the time the page renders and the time the bar snaps to the correct size... not the best user experience.
If you were to disable the OOTB scrolling behavior, then things may get easier, but IMHO realistically you need to rework quite a bit of the default DOM structure to make a 100% dynamic quick launch a viable option. There's not a (great) solution to it without doing a fair amount of rework. You're not going to find a couple lines of magic CSS to pull it off--at least not without disabling the native SharePoint scrolling.