"Minecraft" is damaged and can't be opened on Mountain Lion
The problem actually was that Mountain Lion gives this nonsensical error message to any app not authorized by Apple or from the App Store, by default. You just have to go to Apple -> System Preferences... -> Security &amp; Privacy -> Allow applications downloaded from:

Then select "Anywhere". Seems like this is useful, but I couldn't get Minecraft to run without disabling this security check.