This is a general method for making calls to Android APIs in libgdx. I don't think the Facebook app actually has a "share" activity you can go to without implementing the Facebook SDK in your app, so I'll just do a simple example with Facebook links.
Create an interface like this:
public interface FacebookLinkHandler{
    public void openFacebookPage(String facebookAppURI, String facebookWebURL);
}
Then in your Game subclass, add one of these interfaces as a constructor parameter:
public class MyGame extends Game{
    //...
    FacebookLinkHandler mFacebookLinkHandler;
    public MyGame(..., FacebookLinkHandler facebookLinkHandler){
        //...
        mFacebookLinkHandler = facebookLinkHandler;
    }
    //And when you want to go to the link:
    mFacebookLinkHandler.openFacebookPage(
        "fb://profile/4", "https://www.facebook.com/zuck");
}
Then in your Desktop Application, you can create a version that uses only the fallback:
public class DesktopFacebookLinkHandler implements FacebookLinkHandler{
    public void openFacebookPage(String facebookAppURI, String facebookWebURL){
        Gdx.net.openURI(facebookWebURL);
    }
}
and pass one into your MyGame constructor in the DesktopStarter class. And then in the Android Project, you do the alternative:
public class AndroidFacebookLinkHandler implements FacebookLinkHandler{
    Activity mActivity;
    public AndroidFacebookLinkHandler(Activity activity){
        mActivity = activity;
    }
    public void openFacebookPage(String facebookAppURI, String facebookWebURL){
        Intent intent = null;        
        try {
            mActivity.getPackageManager().getPackageInfo("com.facebook.katana", 0);
            intent = new Intent(Intent.ACTION_VIEW, Uri.parse(facebookAppURI));
        } catch (NameNotFoundException e) {
            intent = new Intent(Intent.ACTION_VIEW, facebookWebURL));
        }
        mActivity.startActivity(intent);
    }
}
and do the same thing. Pass an instance of this into your MyGame class when you instantiate it in your AndroidStarter class.