This is more of an Azure feature question.
You can not stop an Azure Resource Group, because technically the resource group isn't running, but there is also no feature to disable or pause all of the resources in Azure currently.
Like you said, you can pause the Web App roles, however, pausing them, does not remove all the costs of running those web app roles.
There is an open suggestion here: https://feedback.azure.com/forums/34192--general-feedback/suggestions/17670613-hibernate-pause-a-resource-group-or-subscription related to adding in this support.
My suggestion for now, until Microsoft adds this feature (if they ever do), is to just delete the resource group. If you used an ARM Template to create the resource group and the roles within that, than use the same ARM Template to recreate it. If you didn't use an ARM template, than I would recommend using one of the existing Sitecore ARM templates and/or modifying it to meet your needs.