How to make alternative app markets perform automatic updates? I have tried F-Droid and the Amazon App Store, but I had to confirm every update manually. Is there a way to make any alternative app markets (i.e. not Google Play Store) auto-update in the background, without prompting me?
None of them other stores will be able to auto-update your apps unless you are rooted.
Other apps cannot install new apps without root access.

If you do have root access,
1Mobile(1mobile.com) store has this feature. Mobogenie is out there, too, but it has more ads.