Normally I would refrain from commenting on another person's answer, but this action was the first thing I looked at when the question was raised.
As near as I can tell, here was happened in Operation Praying Mantis, on Apr 14 1988. There were three Iranian surface combatants sunk - Joshan, Sahand and Sabalan. Joshan was sunk by Standard MR-1 missiles in the surface to surface mode from Wainwright (CG-28) and Simpson (FFG-56), and Sabalan and Sahand were sunk by bombs from A-6s from Enterprise. That part is referenced from "Electronic Greyhounds", by Michael C. Potter, and it fits with my memory.
There was much, much 5" ammo expended during the operation, which was all from the DesRon 9 ships firing on the Iranian oil platforms with reckless disregard of the fact that they had zero resupply of 5" ammunition in theater. The Commodore encouraged his commanding officers to continue firing at the platforms in an effort to "destroy" them. This is referenced from my experience at the Ordnance Logistics office stateside, where resupply of 5" ammunition needed to be sent out to the Indian Ocean immediately from the West Coast.
And neither the missile fire or the bombs on the Iranian ships, nor the gunfire on the Iranian oil platform constitutes an answer to this question, so I did not offer this operation as a solution to the question. I'm still interested in hearing what the last sinking by gunfire was.