Docker downloads all updates into a directory in /Users/$USER/Library/Caches/com.docker.docker/org.sparkle-project.Sparkle
Let's make this directory inaccessible to the user under which Docker is going to run (you).
Switch to root user:
sudo su
Change directory ownership and prevent access to other users:
cd /Users/$USER/Library/Caches/com.docker.docker/
chown -R root org.sparkle-project.Sparkle
chmod -R 700 org.sparkle-project.Sparkle
From now on Docker won't be able to download or use already downloaded updates in that directory which you can verify by checking the docker logs (Console app):
Failed update installation:
error 20:36:16.153871+0200 Docker Searched /Users/zuse/Library/Caches/com.docker.docker/org.sparkle-project.Sparkle/PersistentDownloads/icnqy76Ox/Docker 63152 for Docker.(app|pkg)
error 20:36:16.153909+0200 Docker No suitable install is found in the update. The update will be rejected.
error 20:36:16.156157+0200 Docker Error: An error occurred while extracting the archive. Please try again later. The update is improperly signed. (URL (null))
Failed update download:
error 20:39:37.867423+0200 Docker Failed to create directory with intermediate components at /Users/zuse/Library/Caches/com.docker.docker/org.sparkle-project.Sparkle/PersistentDownloads with error Error Domain=NSCocoaErrorDomain Code=513 "You don’t have permission to save the file “PersistentDownloads” in the folder “org.sparkle-project.Sparkle”." UserInfo={NSFilePath=/Users/zuse/Library/Caches/com.docker.docker/org.sparkle-project.Sparkle/PersistentDownloads, NSUnderlyingError=0x60000228be70 {Error Domain=NSPOSIXErrorDomain Code=13 "Permission denied"}}
error 20:39:37.869853+0200 Docker Error: An error occurred while downloading the update. Please try again later. (null) (URL https://desktop.docker.com/mac/stable/arm64/63152/Docker.dmg)
error 20:39:37.869883+0200 Docker Error: Can't make a temporary directory for the update download at (null). (null) (URL (null))
Set hosts "127.0.0.1 desktop.docker.com" to prevent docker check update