The hardest part to figure all of this out was the container-selinux requirement. Just find the latest version in http://mirror.centos.org/centos/7/extras/x86_64/Packages/ and install that first. In addition EC2 instances may not have a proper entropy generator so haveged may need to be installed.

The rest is taken from https://docs.docker.com/install/linux/docker-ce/centos/ with the addition of haveged and firewalld. All these have to be done as root so sudo appropriately.

yum install -q -y http://mirror.centos.org/centos/7/extras/x86_64/Packages/container-selinux-2.42-1.gitad8f0f7.el7.noarch.rpm
yum install -q -y http://dl.fedoraproject.org/pub/epel/7/x86_64/Packages/h/haveged-1.9.1-1.el7.x86_64.rpm
yum-config-manager --add-repo https://download.docker.com/linux/centos/docker-ce.repo
yum install -q -y firewalld docker-ce
systemctl enable firewalld
systemctl start firewalld
firewall-cmd --add-port=2377/tcp --permanent
firewall-cmd --add-port=2376/tcp --permanent
firewall-cmd --add-port=7946/tcp --permanent
firewall-cmd --add-port=7946/udp --permanent
firewall-cmd --add-port=4789/udp --permanent
firewall-cmd --zone=public --permanent --add-masquerade
firewall-cmd --reload
systemctl enable haveged
systemctl start haveged
systemctl enable docker
systemctl start docker
setenforce 1

Enable SELinux by modifying /etc/sysconfig/selinux to be

SELINUX=enforcing
SELINUXTYPE=targeted

Then reboot your instance by issuing shutdown -r now

Executing sudo docker version should yield as of the time of this posting...

Client:
 Version:       18.03.0-ce
 API version:   1.37
 Go version:    go1.9.4
 Git commit:    0520e24
 Built: Wed Mar 21 23:09:15 2018
 OS/Arch:       linux/amd64
 Experimental:  false
 Orchestrator:  swarm

Server:
 Engine:
  Version:      18.03.0-ce
  API version:  1.37 (minimum version 1.12)
  Go version:   go1.9.4
  Git commit:   0520e24
  Built:        Wed Mar 21 23:13:03 2018
  OS/Arch:      linux/amd64
  Experimental: false