There are various other Android app store apart from the official play Store (Amazon, F-droid, etc). Any app can request permission to install other apps silently, or may be even a newer version of itself. Play Store uses no private API which is not available to other Android developer. Ask your users to download the apk from your website, if Google Play store won't carry your app, and then built-in a mechanism to auto-update at will.

Yes, there's a permission to install apps silently (android.permission.INSTALL_PACKAGES), but this isn't a permission that can be granted to a third party app. Other bundled app stores can if they're signed appropriately, or if the device is rooted.

Pretty sure even Facebook used this for a few days when they had their auto-updating beta channel. Nobody seemed to really mind.