Is there built-in support for enabling SSL on Azure Container Instances?

There is nothing built-in today. You need to load the certs into the container and terminate SSL there. Soon, we will enable support for ACI containers to join an Azure virtual network, at which point you could front your containers with Azure Application Gateway and terminate SSL there.

As said above, no support today for built-in SSL when using ACI. I'm using Azure Application Gateway to publish my container endpoint using the HTTP-to-HTTPS bridge. This way, App Gateway needs a regular HTTPS cert (and you can use whichever model works best for you as long as you can introduce a .PFX file during provisioning or later during configuratiorn) and it will then use HTTP to talk to your (internally facing) ACI-based container. This approach becomes more secure if you bind your ACI-based container to a VNET and restrict traffic from elsewhere.

To use SSL within the ACI-container you'd need to introduce your certification while provisioning the container, and then somehow automate certificate expiration and renewal. As this is not supported in a reasonable way, I chose to use the App Gateway to resolve this. You could also use API Management but that is obviously slightly more expensive and introduces a lot more moving parts.

I blogged about this configuration here and the repo with provisioning scripts is here.