AWS ACM provides an API to export the public certificate via CLI.
Sadly this is not the case. The link is for AWS ACM Private CA which is related to AWS ACM, but its not the same:
Exports a private certificate issued by a private certificate authority (CA) for use anywhere.
In short, AWS ACM
certificates can't be exported, while those from AWS ACM Private CA
can. However, SSL certs from AWS ACM Private CA
are not public, can't be used on the internet and are not free.
If you require to have control of a private key from your SSL cert, you need to get it from a third party, not from ACM.
I assume that you want private key, not public. Public key you can get from a browser when you connect to your website protected by SSL cert from ACM.