I thought it would be useful to group interoperability into three divisions: Language interoperability, Platform interoperability, and Standards interoperability. While Java has its defining strength in platform interoperability, C# has it's strength in language interoperability. Both have strengths and weaknesses in standards interoperability.
This is the level and ease of integration with other languages. Both the Java Virtual Machine and the Common Language Runtime allow you to write code in many different languages, so long as they compile to byte code or IL code respectively. However, the .NET platform has done much more than just allow other languages to be compiled to IL code. NET allows multiple languages to freely share and extend each others libraries to a great extent. For instance, an Eiffel or Visual Basic programmer could import a C# class, override a virtual method of that class, and the C# object would now use the Visual Basic method (polymorphism). In case you were wondering, VB.NET has been massively upgraded (at the expense of compatibility with VB6) to have modern object oriented features.
Languages written for .NET will generally plug into the Visual Studio.NET environment and use the same RAD frameworks if needed, thus overcoming the "second rate citizen" effect of using another language.
C# provides P/Invoke, which is a much simpler (no-dlls) way to interact with C code than Java's JNI. This feature is very similar to J/Direct, which is a feature of Microsoft Visual J++.
Generally this means OS interoperability, but over the last few years the internet browser has emerged as a platform in itself.
C# code runs in a managed execution environment, which is the most important technological step to making C# run on different operating systems. However, some of the .NET libraries are based on Windows, particularly the WinForms library which depends on the nitty gritty details of the Windows API. There is a project to port the Windows API to Unix systems, but this isn't here now and Microsoft have not given any firm indication of their intentions in this area.
However, Microsoft hasn't ignored platform interoperability. The .NET libraries provide extensive capabilities to write HTML/DHTML solutions. For solutions which can be implemented with a HTML/DHTML client, C#/.NET is a good choice. For cross-platform projects which require a more complex client interface, Java is a good choice. Kylix, a version of Delphi which allows the same code to compile to both Windows and Linux may also be a good choice for rich cross-platform solutions in the future.
Microsoft has submitted the C# specification as well as parts of the .NET specification to the ECMA standards body.
These are all the standards like databases systems, graphics libraries, internet protocols, and object communication standards like COM and CORBA, that the language can access. Since Microsoft owns or plays a big role in defining many of these standards, they are in a very good position to support them. They of course have business motivations (I'm not saying they are or are not justified) to provide less support for standards which compete with their own - for instance - CORBA competes with COM and OpenGL competes with DirectX. Similarly, Sun's business motivations (again I'm not saying they are or are not justified) means Java doesn't provide as good support for Microsoft standards as it could.
C# objects, since they are implemented as .NET objects, are automatically exposed as COM objects. C# thus has the ability to expose COM objects as well as to use COM objects. This will allow the huge base of COM code to be integrate with C# projects. .NET is a framework which can eventually replace COM - but there is so much deployed COM code that by the time this happens I'm sure .NET will be replaced by the next wave of technology. Anyway, expect .NET to have a long and interesting history!
I hope this has given you a feel for where C# stands in relation to Java and C++. Overall, I believe C# provides greater expressiveness and is more suited to writing performance-critical code than Java, while sharing Java's elegance and simplicity, which makes both much more appealing than C++.
The Microsoft Developer .NET homepage can be found at http://msdn.microsoft.com/net