Is there a cross-OS GUI framework that supports embedding HTML pages?
It is strange that Qt is not for you. You may be surprised to hear Sencha's Architect and Animator products use Qt and QWebView for cross platform JavaScript applications with full menus and icons and executables and system dialog boxes and file I/O.
It currently works Windows, OSX, and Linux. 
They use an in-house developed library called ion to load and interact a JavaScript application. They provide some helper classes for JS to use.
A simple skeleton c++ application which uses Qt to create and load a window and create a web view in that window and load html and other content from file into that view.
Another solution is Adobe's Air which is like a browser with native support. It also provides deployment.
I want to develop a desktop app to be used cross-system (win, mac, linux), is there a GUI framework that would allow me to write code once for all 3 platforms and have a fully-scriptable embedded web component?
I need it to have an API to communicate between app and webpage javascript.
I know C#, JavaScript and a little bit of python.