It would be for a disciplinary panel to decide - after considering the broader context, the implications, the potential for premeditation and intent and any reasonable excuse / explanation of the employee - whether, or not, a disciplinary process should be adopted.
There will be a sliding scale which can only realistically be applied AFTER proper consideration of the circumstances.
If it were a repeat, frequent, intentional misdirection and misrepresentation of the circumstances in order to achieve an unreasonable advantage for the perpetrator then yes - gross misconduct might be an appropriate finding.
If it were a clumsy, accidental, single incident of IT fat fingeredness, then perhaps a quiet word would be more appropriate possibly some additional targeted training or mentoring.
At the very least - the person responsible for closing the meeting and terminating the video record has failed in their duty to do so.
The person that has uploaded the video record has then compounded the error which will need to be corrected anyway.
There is a case for damaging PC reputation and credibility - if I had been 'asked to leave the public session' (and I presume you meant there was a motion on the agenda which specified the reason for the closed session and that there was a vote of the PC before the public were asked to leave?) and then I watched any part of the session I had been asked to leave on a public broadcast YouTube video - I would be properly miffed....
Yes, you might have a circumstance which warrants disciplinary investigation - if the broader context supports it.
Have you put your case to the chair / HR committee for consideration??