I, too, would like to see this functionality -- as buggy as it may currently be -- in gnome-shell. It's pretty fancy. But, regardless of how cool it is I will never switch back to Unity.

After a cursory glance at the facilities that enable the Webapps interface to work, it appears this is not possible - at least at the moment.

I considered perhaps making a gnome-shell extension to create an interface object that would act as the Unity object in gnome-shell. This couldn't possibly work though because the facility to install the webapps is a Firefox/Chromium extension called Unity Desktop Integration. This extension allows for installations to be scripted on a web page.

Gnome-shell, too, has a plugin called Gnome Shell Integration which allows extensions to be installed from a web page, however it is only available for use on extensions.gnome.org for security reasons. So other sites cannot use the scripting available on extensions.gnome.org on another site (such as Pandora, Gmail, etc).

So, unless somebody makes a bridge between the Javascript in the browser and the Javascript in gnome-shell it is currently impossible. Again, it is a potential security vulnerability to allow for any arbitrary site to have script access to gnome-shell, so even if a new FF/Chromium extension were to pop up translating the Unity Webapps capability to gnome-shell it would have to be restricted to a limited subset of sites. Not saying this wouldn't still be useful.

Also, Unity webapps are launched using a program called unity-webapps-runner. Therefore, somebody would have to create a similar program (or modify Unity's) to actually run the webapps in gnome-shell. Then again, I don't believe that Unity is scriptable like gnome-shell (ie. gnome-shell extensions), so they may need this program to make the webapps show up in the unity notification panel. This type of facility already exists in gnome-shell in the form of extensions, so this step may not be necessary.