I can no longer install apps (apk's) from the Amazon App Store, or from downloaded apk files, but the market works fine.
I am not rooted, and I am still running 3.2 (US). This started in the last few days, but I can't think of anything that I have done.
I cold booted, but I still get the same results; The install dialog box comes up with the options at the bottom "Install" or "Cancel", however, install does nothing, the window is active because you can press "Show All", or "Cancel" and those do work, but not "Install". I can't think of anything that I have overlooked.
The most a) likely solution comes in the form of b) security applications interfering with the installation, c) package incompatibility with the device or screen filter applications used to lower the device's brightness below the minimal limit of the Brightness settings.
In my case it was the application "Lux" which was to blame. Thus make sure to untick "stay alive in the "generals" option menu. And kill the Lux app. Now installation proceeds unhindered. Avast was not at fault. So don't freak,- often the solutions are trivial - finding the answers isn't.
Thanks to anyone who contributed in the original thread and to xda.