Overly Pious Gedalia
The Fast of Gedalia has nothing to do with the High Holidays, other than the fact that all fast days are a call for repentance. It is more directly related to the events connected with the destruction of the Temple.
It is referred to in the Tanach as fast of the seventh month, which is Tishrei. The assassination of Gedalia Ben Achikam at the hands of another Jew, Yishmael Ben Netanyah, took place on Rosh Hashanah. The fast is always on the day after.
The King of Ammon actually hired Yishmael to kill Gedalia. He was upset that many Jews who had fled to Ammon, returned to Israel to be part of the remnant that remained after the destruction of the first Temple by Nebuchadnezzar. This tragic murder put an end to any hope of the Temple being rebuilt, at that time.
The Messilat Yesharim uses this incident of Gedalia’s assassination to teach that a person should not be overly righteous. Gedalia was warned of the plot to overthrow him. He “piously” chose not to believe such terrible Lashon Hara, and did not take measures to protect himself. He paid an unnecessarily heavy price for his foolishness. There are times when it is good to give the benefit of the doubt. But there are others, when self righteousness is potentially dangerous. Gedalia paid with his life for trading good sense with being pious.