I'm puzzled by that error, I think that's the non-java 'app' thing; if you've downloaded and installed the java version - which should have been 'MinecraftInstaller.msi" - that should have created a shortcut icon, and I don't think that can give that error message. Sorry, I'm just confused about what's going on, I've never seen that message before at all.
There is just one other idea, that I can think of; go to https://minecraft.net/ and after logging in, go to 'Downloads', but click on 'Download for Linux" instead of the Windows one. You should get a file called Minecraft.jar - try running that. It's exactly the same as the windows download, it's just getting the file directly instead of using the "Microsoft Installer" thing.