AFAIK there is no way to externally view the call stack. Once inside a method you can get the call stack thus:
StackTraceElement[] stackTraceElements = Thread.currentThread().getStackTrace();
I am not sure this will help you as you wish to check that all methods are called.
I would suggest a two pronged approach to testing; this is something I have employed and works.
Firstly unit test. This should be 'white box', the tester knows what is being called and creates tests to cover all the paths; or as many as are practical.
Secondly integration test. This is a 'black box' test, validate that the software is delivering what is expected. This will often be from the UI point of view and can be automated for reliable reproduction.
In addition to this, if the application goes through several releases, is regression testing. Ensure the application still do what it was doing prior to any changes.