The only thing you need to host sphinx documentation is a static file server (the search works without a back end, see my answer here.
That said, using a private readthedocs server is probably over-engineering.
Just deploy the files to a static file server and point the base URL (e.g. docs.myapp.com) to the index.html file.
You can automate the deployment with git hooks.
For the sake of completeness: I am sure it is possible to get a local readthedocs server to build your project. But readthedocs is explicitly not designed for On Premise deployments and you might find it hard to get professional support. I was involved in a scenario where the Dev Ops team decided it’s way easier to automate the deployment using their usual set of tools after we struggled with build/performance issues of our local readthedocs instance.