Does X-Windows have to be installed on a Linux-box in order for Java to display fullscreen graphics?
Well "fullscreen graphics" is a bit vague.
Anyway, apparently there is a an effort ongoing to access the framebuffer from Java: Framebuffer Toolkit.
The objective of this project is to produce a body of code which is a lightweight framebuffer-based peer implementation for AWT and Swing. The goal of this code is to remove the dependency on X or other graphics layers such that graphics can be redirected to a framebuffer (e.g. a raw buffer, VNC, etc.). This example implementation will prefer pure-Java solutions, with public extension points available to enter native resources as necessary.
To really display something graphical on the screen, yes. Bud there is a headless version of the JRE for just running it. You won't see any graphical output, but it will run. Alternatively, you can log in remotely and use X forwarding to run the java code on the server but let the client handle displaying graphics.