The USS Constitution, launched in 1797, is the only remaining ship in the US Navy to have sunk an enemy vessel

From The Navy needs to stay focused — or we’re doomed by Glenn H. Reynolds.  I don't know if the claim is true or not.  I cannot readily produce any refutation.  Famously the Battle of Coral Sea in WWII was the first naval battle in which neither fleet saw or fired on one another.  All the sinkings and damage were from aircraft on aircraft carriers.  Midway, just a month later was similarly conducted without visual contact or direct fire.

But subsequent naval battles such as  Iron Bottom Sound, the Battle of Philippine Sea still remained traditional fight between the two enemy fleets, slugging it out.

However, no US Navy ships of WWII remain in commission.  

In the Korean War there was only a single surface engagement but three North Korean torpedo boats were sunk.  The only US ship in that engagement was the USS Juneau, decommissioned in 1955.  

Similarly there was an encounter between the US Navy and some North Vietnam warships during the Vietnam War but all those ships have been decommissioned.  I can not think of any other surface engagements between capital ships since then involving the US Navy so I suspect Reynolds is correct.  

Here is what caught my attention in his article:

Last week I toured the USS Constitution, in Boston Harbor. Launched in 1797, “Old Ironsides” is the oldest commissioned warship afloat.

It’s also the only remaining ship in the US Navy to have sunk an enemy vessel. That may explain some news that came out this week.

The substance of the article is also worthwhile reading.