Java-Based Mac App
The answer by l'L'l is not entirely correct. You do not need to translate your Java code to Objective-C. You can ship a Java-based app in the Apple Mac App Store.
Apple changed their policies to now allow Java-based apps for sale in the Mac App Store. This applies only to Mac apps, as far as I know, not for iOS. There is no Java implementation released that runs on iOS as far as I know. Sun demonstrated a prototype once publicly, but never released anything for iOS afaik.
See my answer to a similar question. The successful app Moneydance is an example, as noted on this answer.
Nutshell
- You create a thin Objective-C app that wraps your Java app, using Xcode.
Other people have provided templates for this minimal code.
- Your Mac app must include a JVM (Java Virtual Machine).
In other words, your Mac app cannot access a JVM that may already be installed on the user’s Mac.
- You may use Swing or JavaFX for presenting a user interface.
Both Apple’s and Oracle’s implementations of Java ship with extensions to Swing for a Mac look and feel. The latest implementations from Oracle also include JavaFX.
Oracle provides these instructions on how to package your app for the Mac App Store.
GameCenter
I do not know the details of accessing GameCenter from the Java-based part of your game. You may need to write some Objective-C in order to interact with Apple's GameCenter libraries/framework.
As noted by MadProgrammer (comment on question), your Java code may interact with Objective-C code by using either:
Xojo
By the way, another non-Objective-C platform for shipping apps in the Mac App Store is Xojo, formerly known as RealBasic. A very slick set of tools with a fully object-oriented programming language. Popular with both beginning programmers and professionals.