I'd rather that joke were not made. There is a difference though between a discreet 'not a great moment' and public 'absolutely inacceptable [and we should hound the poor sap out of all gainful employment opportunities for life]'. A classic management principle is 'praise in public, criticize in private'. While the joke was was indeed inappropriate, the public reaction was 10x worse.
I can agree that there should be no public flogging, provided it is first couple times. If it has been explained to the individual that certain behaviour is considered unprofessional and yet they insist on engaging in it then... well, there should be consequences.