Can Ionic Framework run in desktop web browser like Chrome, Mozilla, IE9+?

If you just want to test, check this out:
http://ionicframework.com/docs/guide/testing.html

Desktop browser testing

Testing your app in a browser is as simple as running the serve command in your project's root folder.

$ ionic serve

It sounds like the Ionic team is making this a priority now:

As many of you may have seen, Progressive Web Apps are going to be a major focus of Ionic's going forward. We want to help Ionic developers ship great apps to the App Stores, but also take the same code and deploy them to the web to maximize distribution and discoverability. We think this is a unique strength of Ionic that native SDKs are missing today, and want to make sure Ionic developers can maximize that benefit.

(source)

Ionic recently added a browser build target that generates a web app. If you run ionic build browser, you'll see that Ionic creates a www folder containing assets you can deploy to a web server.

There's a GitHub issue that links to some documentation, particularly a tutorial on how to deploy to Firebase that describes the browser build process (as well as Firebase-specific deployment steps, as you might imagine).

UPDATE : As mentioned by @RationalDev, Ionic now supports browsers in general. However, they haven't explain what browsers.

Ionic 1.2 marks the first release where we actively recommend Ionic for those looking to build a mobile website

http://blog.ionic.io/announcing-ionic-1-2/

=========================================================================== As mentioned in the Browser Support section of the Ionic Framework documentation, Ionic is really geared towards hybrid mobile apps.

Ionic is focused on building native/hybrid mobile apps rather than mobile websites.

As such, our browser support tends to be whatever Web View API is available to native apps on a given platform. For Ionic 1.0.0-beta.6 "darmstadtium-dingo", that means UIWebView for iOS 6+, and Android 4+ (with some support for 2.3). Windows Phone and FirefoxOS support is on our roadmap.

However, it does work just fine in most WebKit browsers (Chrome and Safari in particular). In fact, I'd say most developers develop their app in Chrome 90% of the time and then test on their devices or a simulator.

I do know that FireFox is simply not supported. I don't know how well Ionic works in IE X.

If you want a mobile app, Ionic is a great option. However, if you are really targeting the desktop, I'd say you need to look elsewhere. Ionic provides no guarantee of support for desktops browsers and it's interface would limit the more feature rich and spacious capabilities of a desktop browser.

For AngularJS, you might want to look into AngularUI and AngularUI Bootstrap in particular. It has a really diverse desktop oriented feature set.