If you want to automate or install Docker Desktop on Mac using Terminal Command you can make use of the Docker Cask command.
Docker Desktop installation logs:% brew install --cask docker
...
==> Auto-updated Homebrew!
==> Tapping homebrew/cask
Cloning into '/opt/homebrew/Library/Taps/homebrew/homebrew-cask'...
...
Receiving objects: 92% (627470/682032), 312.41 MiB | 1.77 MiB/s
Resolving deltas: 100% (484134/484134), done.
Tapped 4106 casks (4,178 files, 345MB).
...
==> Downloading https://desktop.docker.com/mac/main/arm64/89412/Docker.dmg
==> Installing Cask docker
==> Moving App 'Docker.app' to '/Applications/Docker.app'
==> Linking Binary 'docker-compose.bash-completion' to '/opt/homebrew/etc/bash_c
...
==> Linking Binary 'docker.bash-completion' to '/opt/homebrew/etc/bash_completio
🍺 docker was successfully installed!
Not just type Docker in Spotlight Search and you should be able to find it.
The first usage will give you a message "Docker Desktop needs privileged access." Docker Desktop needs privileged access to install its networking components and links to the Docker apps. You will be asked for your password.

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