I am currently teaching a student on how to code a game in Java on a Mac. GameCenter is a service provided by Apple that lets one see their game stats on a leaderboard and lets them play online. Does anyone know if it's possible to implement GameCenter in Java or if it's even possible to publish a Java app on the Mac App Store?
You would need to translate parts of your Java application over to Objective-C, which is what OS X and iOS support.
Fortunately in the last couple of years there has been a decent amount of progress and attention in terms of tools, techniques, and methods that can help you do this. There's even an open-source tool/runtime that can take Java code and turn it into Objective-C:
It's fairly popular, and can probably set you in the right direction.
EDIT: I've added the word parts above since it was apparently unclear to Basil Bourque, and/or possibly anyone else who may have taken the word translate as having to re-write your entire app in objective-c.