You make a report in writing if you could personally be held liable, and you want your disagreement on the record. But doing so is a burning your bridges moment.
Not quite sure how you handle this if you are remote...
I think this is dangerous. If push comes to shove, you will not have anything to defend yourself with, because of no documentation.
Normally this discussion arises before the bad thing has been done, or if you weren't involved and found out about it. If you in fact committed it on their behalf without telling them first, then you are actually the bad guy.
Things are much more likely to go sideways if (a) there was already some background feeling that this person had to go, or (b) the person makes a nuisance of themselves after legal or higher-ups have already decided that no, this was fine.
Seems to me when a manager "jumps straight to firing" someone, either they are insane or there is something missing to the story. Insane is very much not impossible (and disfunction itself obviously) but missing story is not impossible either.