The measure would have placed a charter amendment before voters in November, asking whether the executive should have to get the council’s approval to fire the sheriff.
Republican councilman Dave Greenspan said the measure would have strengthened the independence of the sheriff’s investigations.
"That position needs to be totally separate and protected from removal in the event that an investigation touches too closely to an executive," he said.
But Democratic Councilman Dale Miller said during the meeting the move would make the relationship between the two offices unclear.
Greenspan said the impetus for his proposal was County Executive Ed FitzGerald’s 2013 firing of sheriff Bob Reid. He said the council received no advance notice and still doesn’t know the reason for the firing.