No, you need not push the image to ACR first, just let the image stay in the docker hub. For example, deploy the Nginx docker image to ACI, the Azure CLI command like below:
az container create -g resourceGroup -n aciName --image nginx --ports 80
As the command shows, you can use the docker image. Actually, the docker hub is the default registry. When you use another registry, you need to add the parameters --registry-login-server, --registry-username and --registry-password. For more details, see az container create.
It also shows clearly in the Azure portal, when you create ACI in the portal, you can see it like below:
