Can I get an image digest without downloading the image?

For newer versions of Docker, the inspect command provides the correct value:

docker inspect --format='{{index .RepoDigests 0}}' waisbrot/wait

For older versions, fetch the value from the repository following this example with the main Docker repo:

curl -H "Accept: application/vnd.docker.distribution.manifest.v2+json" \
     -H "Authorization: Basic ${username_password_base64}" \
     'https://auth.docker.io/token?service=registry.docker.io&scope=repository:waisbrot/wait:pull' 

Naive attempts to fetch that value fail because the default content-type being selected by the server is application/vnd.docker.distribution.manifest.v1+prettyjws (a v1 manifest) and you need to v2 manifest. Therefore, you need to set the Accept header to application/vnd.docker.distribution.manifest.v2+json.

This is how you do it today using a V2 manifest.

docker manifest inspect <REMOTE IMAGE>:<TAG> -v

Your output is JSON:

{
  ...
  "Descriptor": {
        "mediaType": "application/vnd.docker.distribution.manifest.v2+json",
        "digest": "sha256:d13e102941a9f7bd417440f62f9cb29de35f6acb13a26cbf6a34f4c7340f0b63",
        "size": 3255,
        "platform": {
            "architecture": "amd64",
            "os": "linux"
        }
  },
  ...
}