How do I add a CA root certificate inside a docker image?

The task itself is not specific to docker as you would need to add that CA on a normal system too. There is an answer on the askubuntu community on how to do this.

So in a Dockerfile you would do the following (don't forget chmod in case you're running the container with a user other than root):

ADD your_ca_root.crt /usr/local/share/ca-certificates/foo.crt
RUN chmod 644 /usr/local/share/ca-certificates/foo.crt && update-ca-certificates

To simplify/standardise all container builds, we now host our certificates on a central HTTPS server and build them into our containers like this:

# Debian stretch based container
RUN curl -ks 'https://cert.host.server/ssl_certs/EnterpriseRootCA.crt' -o '/usr/local/share/ca-certificates/EnterpriseRootCA.crt'
RUN /usr/sbin/update-ca-certificates

Alpine-based containers don't have the tools immediately available so require a bit more work to achieve the same:

# Alpine based containers
RUN apk update && apk add curl
WORKDIR /usr/local/share/ca-certificates
RUN curl -ks 'https://cert.host.server/ssl_certs/EnterpriseRootCA.crt' -o '/usr/local/share/ca-certificates/EnterpriseRootCA.crt'
RUN /usr/sbin/update-ca-certificates

If you also want to update your Java truststore (same as on any computer):

RUN keytool -keystore /usr/lib/jvm/java-8-oracle/jre/lib/security/cacerts -storepass changeit -noprompt -trustcacerts -importcert -alias EnterpriseRootCA -file EnterpriseRootCA.crt

It's also worth noting that this definitely needs to use the .crt extension. I initially tried this with a .pem cert file (I thought they were interchangeable, so others might also), which is not linked by update-ca-certificates.