No. Having Docker installed doesn't allow anyone to use it by default.
$ docker info
Got permission denied while trying to connect to the Docker daemon socket at unix:///var/run/docker.sock: Get http://%2Fvar%2Frun%2Fdocker.sock/v1.39/info: dial unix /var/run/docker.sock: connect: permission denied
As an ordinary user, you don't have the privilege to even connect to the Docker daemon, let alone instruct it to run something in a container.
On the other hand, any user who is in the
docker group effectively has root powers on the host. It's very difficult to allow users to create a Docker container without giving them root powers on the host. You can allow users to run an application in a Docker container if the container is configured securely. A secure configuration must not grant access to any sensitive files on the host and should include a
USER directive that doesn't grant root access inside the container. (Preventing root from escaping a container is possible, but tricky.)
The “sudo replacement” is
adduser alice docker or having
alice ALL = ( : docker) docker in
/etc/sudoers, not merely installing