Get your story out first.
I would have already done a company wide email on the heels of his "reply all" and used it as a teaching moment of what not to do and the expectation for email communications.
Example: Hey everyone, recently a mass email went out to the company describing a serious problem with a new policy. While that matter is being addressed through more proper channels, I want to take this real time situation to remind or teach each of you several things regarding corporate policy and email use. First, let's talk email. Great rules of advice are "never email angry" or reply quickly. Write the email and save it as a draft. Let it sit for a few hours or overnight. Sleep on it. Then read it again, read it out loud. If you still feel the same then it is time to send. However, as a courtesy you should also schedule a face to face with your immediate supervisor or higher if the situation warrants.
This brings us to a second reminder, chain of command. The entire staff is never your chain of command. You have supervisors, managers, and so on. It is entirely acceptable and welcome to air challenges with the appropriate levels above your pay grade. Report issues or problems within the proper chain of command, wait an appropriate amount of time for action or resolution and then follow up. Document your follow up visits in an email, I have zero issue with that. We here manage both up and down, that means you the staff have just as much right to hold management accountable as they do of you. Does anyone not fully understand the chain of command? I will be happy to schedule face time to review if needed. Additionally, I have attached contact info of all the supervisors, managers, etc and this info can also be found in your XXXX.
Just to recap, you should never abuse the "reply all" function to air issues. You should be comfortable approaching and or emailing your supervisors to address issues. We absolutely will listen and reply or act if needed. More importantly, we will close the communication loop with you so you are aware of whether the issue was handled or not (respecting privacy if situation demands).
If anyone has any further questions or concerns, please see me or call me....XXXXXXX
***I have been there done that. The reason I suggest the above is you are ahead of the issue and not hiding it. Sweeping under the rug days are long gone. I strongly recommend this action because believe it or not, you will have staff supporting YOUR decision when you terminate said employee. While some may laugh at the audaciousness of said employee, I guarantee many of the staff are aghast.
So you professionally address it now before taking any actions. You recommunicate what is acceptable. You reach out to all staff. Then when actions take place, YOUR version is already out and no matter what he says afterwards (in case you keep him on board) his comments become irrelevant and his peers dismiss his words since they now know what happened straight from mgmt.
I assure you, it is tough changing corporate culture to this type of mindset as it has not been a common practice. Once it is implemented though, the rewards are endless. I had the luxury of founding my company on a mindset like this so the staff never knew any different. However, I am now neck deep in a **it show that relied heavily on reactive and punitive practices.