According to the US Naval Institute, less than 30 ships of the US Navy have been lost since the end of World War II.

Five of those occurred during Korea, the last of which was the USS Sarsi, which sunk on August 27th, 1952 due to a mine. The other losses in Korea were due to mines as well. As far as I can tell, the USS Sarsi was the last time a US Navy ship was destroyed in war by the enemy, but you have to go back to World War II for a loss due to enemy action. The next closest might be the USS Liberty, but she didn't sink, but was merely scrapped later as the damage was beyond repair.

According to the Naval Institute list, the last time a US Navy ship was lost at sea, was the USS Frank E. Evans, which split in half on June 3rd, 1969 following a collision with an Australian ship.

There have been ships lost since then, but they seem to have all been cased where she ran aground, rather than sunk at sea, such as the USNS Sgt. Jack J. Pendleton on Sept. 23, 1973 and a few more on the near side of the twenty year rule.

I'm somewhat purposefully conflating "abandon ship" with lost at sea for the purpose of this answer, so while I feel it might be doubtful, there may be a case where a ship was abandoned, but not lost. Searching around, I was unable to find anything to show it happening though.

Not sure if this is the most recent example, but I suspect it may be the most recent that doesn't run afoul of the 20-year rule:

A few minutes googling brought me to the story of the US Coast Guard Cutter Mesquite which was replacing a buoy in Lake Superior in December of 1989 when it ran aground on a reef. Ironically, the buoy in question was supposed to warn ships away from the very reef that the Mesquite went aground on.

Attempts were made to refloat the ship, but ultimately proved unsucessful, and the ship was abandoned.