Not to burst your bubble here, but the Java bindings for GTK+ (through the java-gnome project) aren't available for Windows.
Even if it were available on Windows, GTK+ is still a native library, which means that you'll have all the cross-platform incompatibility problems associated with it. Swing is really the only way to go if you're looking for "write-once, run anywhere" pure Java code.
Getting GTK+ working with C++ on Windows should be easier. You probably actually want
gtkmm, which is a C++ interface to GTK+ (GTK+ itself is a plain C API and doesn't take advantage of any of the features in C++, like memory management, polymorphism (the "right" C++ way, I should add), and the Standard C++ library/STL).