There's no reason clients couldn't, as far as I know (just like the various anti-spam measures where clients check for records that indicate what servers are allowed to send mail from a domain and consider it spam if those records don't match), but I also don't know of any clients that actually do. And a CAA record still doesn't actively prevent issuance of a certificate, which is really what would be needed.

There used to be a pinning scheme called HPKP, but the trouble with it is it's very easy to accidentally (or via malicious third-party action) end up in a situation where your web site becomes unreachable by people who've visited it previously because the key changed and the pin prevents their browser recognizing the new key as valid:

https://scotthelme.co.uk/im-giving-up-on-hpkp/