You cannot *download & reinstall/restore* iOS 6 or below on any device that qualifies for iOS 7. iOS 7 will be installed on it.
If the device ALREADY HAS iOS 6 installed on it and you reset the device ON THE DEVICE ITSELF under Settings>General>Reset>Erase All Content and Settings, the iOS version will NOT be updated. It does NOT check for signing when resetting in this way. It will remain at the version currently on the device. If you are on iOS 6.0, that's where it will remain. If you are on iOS 5.1.1, that is where it will remain.
If you restore the device using iTunes, the device WILL be updated to iOS 7 or the highest iOS version that the device is capable of running. This is where the signing of the iOS is checked and this is where it WILL be updated if you are not on iOS 7 or what is current for the device.
Activation does NOT check for iOS signing since ALL devices, regardless of their iOS version need to be activated...iPad 1st Gen, 2nd Gen, iPhone 4. It make NO difference. Activation has nothing to do with the iOS being updated or not updated.