Docker Security Scanning: How to integrate Clair with a docker private registry?
Both Docker Registry and Docker Engine have API interfaces.
I consider it should be possible to implement a Docker plugin for this integration to have images scanned for example as somebody pushes them. (side note - Docker registry is an image registry, not container registry).

This scanner functionality is also possible through the commercial Docker Enterprise edition.
https://docs.docker.com/docker-cloud/builds/image-scan/

UPD here an example with clair/docker, looks quite simple
http://blog.xebia.com/docker-containers-vulnerability-scan-clair/