Does Minecraft use applets or is it just a standalone desktop Java app? Does installing Minecraft create any vulnerability to this bug on a machine whose browsers have deactivated any Java plugin?
Besides downloading the Java app, you can play Minecraft in the browser through an applet at https://minecraft.net/play.
So no, if you are not running Java in the browser, a malicious web page can't run neither Minecraft nor other Java apps that you have locally installed.
