There is an API for Cloud Run:
https://cloud.google.com/run/docs/reference/rest
The gcloud SDK is only an interface to connect to the GCP APIs. If you run any command with the flags `--verbosity=debug` and `--log-http`, you'll see each call the SDK is making.
You can make the calls manually, but it is non-trivial. Nevertheless, all the actions that can be taken in Cloud Run (including deployment), can be taken through the API.