How to build native C++ apps with HTML/CSS UI?

Yes, it is possible. What you want is a C++ web framework. You could start by looking at CppCMS—a Free High Performance Web Development Framework (not a CMS) aimed for Rapid Web Application Development.

Edit: After the question has been clarified, I think Awesomium would be a good fit for your needs.

You can achieve the same functionality by embedding WebKit (or Gecko) but as far as I know that is exactly what Awesomium does behind the scenes.

An alternative is librocket—a C++ interface middleware package designed for game applications. As you can guess from the description it is a good fit for games or real-time applications.

Sciter is exactly that - embeddable HTML/CSS engine with C/C++ API. Compact and multi-platform.

And check this article.

Sciter has the same feature set as WPF but uses HTML/CSS instead of XAML and uses native API.

Features out of the box:

• HTML, CSS, SVG, aPNG (animated PNGs), image sprites,
• <plaintext> - editor with syntax highlighting,
• <htmlarea> - WYSIWYG HTML editor,
• <frame type=pager> - print preview and HTML/CSS print,
• animations,
• HTTP client, REST/JSON client, WebSocket's, DataSockets,
• etc.

The same application with Sciter UI running on Windows, MacOS and Linux from the same sources:

Disclaimer: I am an author of Sciter Engine.

2021 update: you can also use Sciter.JS that is version of Sciter that uses standard JavaScript (ES2020) script engine.