My Kindle Fire HD 7" is sort of Android 4.0.3 (that is what it is based on) and I am able to run a Player that I built and installed via Eclipse.
But I tried downloading and installing the one from the 'My Products' page. When I clicked on the Player to download, it popped up a message saying that it would put the file in the Downloads and, I suppose, that is where I was supposed to install it from. Then I found the file there and clicked on it and got the same "Cannot open file" message.
I could not find any settings to change what could and could not be installed.
Then I repeated the process on my Nexus 7. The download did not automatically install it, I had to go to my Downloads directory and click on it. And it installed fine.
So, I don't know if it is a settings issue in Kindle or the apk is just not compatible.
Cheers,
Ancient Lady
AGK Community Tester and AppGameKit Master