The NVAF did go after ships: In 1972, two MiG-17s attacked the destroyer U.S.S. Higbee (the only ship in the Navy at the time named for a woman) and the guided missile cruiser U.S.S. Oklahoma City. The destroyer took a 550-pound bomb and was moderately damaged, while the cruiser was unharmed. The NVN claimed the destroyer as sunk and the cruiser damaged. Higbee was back on the gun line off NVN within a month (three weeks' repairs at Subic Bay).
In January, 1973, the last MiG kill of the war took place when a MiG-17 tried going after the carrier U.S.S. Midway on Yankee Station. An F-4 from VF-151 shot him down short of the carrier.
note that timing.. during Linebacker II (when the US was going all out against Hanoi and Haiphong). Timing is very important here. I recall the US Navy getting a few SAM kills with Terrier missiles from cruisers during the war as well.