- Apple ID
- Apple Online Store
- App Store
- Bug Reporter/Radar
- iCloud Apps e.g. Mail.
- iTunes Connect
- iTunes Store
- WWDC
Pretty much all of the internal apps will be in Java unless they are using off the shelf software e.g. discussions. In addition you would have all of the companies they acquired and whatever tech stacks they use e.g. Siri.
I have no idea where the idea of Java 6 being in use comes from.
Possibly from confusion with the last time Apple actually shipped a JDK/runtime to the public. They used to produce their own optimized distribution, which had better Cocoa widgets among other things. It was part of the original big push to get decs on OSX, it’s mentioned in Jobs’s keynotes etc. They later deprecated it once they reached critical mass, and discontinued it; if i remember correctly, the last jdk they shipped was a v6.
i.e. it's probably more about fostering great UIs for the end customers rather than curbing Java altogether as a platform.
Somewhere along the way, this part of Apple culture got lost. I think Apple's commercial success is part of the reason. The traditional Mac user base got diluted by lots of newcomers who didn't have the same high expectations and requirements for good and consistent user interfaces. The traditional Mac developers got diluted by programmers from other platforms, who – although not being technically inferior in any way – did not have the same passion or understanding of Apple's UI philosophy, often not even bothering to study the Human Interface Guidelines.
Funny thing is that those are both products that many are reaching for alternatives to whenever viable. As powerful as they may be as tools, there's quite a lot that's throughly mediocre or even bad about them.
There are examples of truly great apps that use their own look and feel, but I don't think either of those are among them.