IOS – How to turn off notifications from the home screen of the iPhone in iOS 9

There is no way to disable push notifications. They are handled by Apple's servers and will be sent to your device even if you uninstall the program.

Disabling all the app's respective settings under the Notification Center will only hide them. Refer to Apple's developer Push documentation for more details.

Specifically, look at the Apple Push Notification (APN) Service, which "transports and routes a notification from a given provider to a given device." Apple says the following about the APN:

Sometimes APNs might attempt to deliver notifications for an application on a device, but the device may repeatedly refuse delivery because there is no target application. This often happens when the user has uninstalled the application. In these cases, APNs informs the provider through a feedback service that the provider connects with. The feedback service maintains a list of devices per application for which there were recent, repeated failed attempts to deliver notifications. The provider should obtain this list of devices and stop sending notifications to them.

—Source

In short, Push notifications are handled outside of the device. To stop them, once a program has been installed, means the developer of the program must remove your device from the list (or the notification must fail enough for the APN to stop sending them entirely). As a user, there is not way to "stop" them, just hide them from within iOS.

You can now simply go to System Preferences > App Store, and turn off "Automatically check for updates."

No messing with firewalls, and just works with App Store. So just remember to check every now and then manually to find out when you have updates!