The XO (Executive Officer) is second in command, and would take over if the CO was killed. This happened in WWII when subs spent more time on the surface, and were subjected to strafing from aircraft, for instance. One Medal of Honor captain ordered his boat to dive while mortally wounded and left topside, I believe.
After the XO, it would be one of the Department Heads (second tour officers such as the Weapons Officer (not always there), the Navigator, or the Engineer); this would be based – as all command succession is – on rank first, followed by seniority. This assumes all officers are Unrestricted Line Officers who are eligible for command at sea; that’s why the Supply Corps Officer on board wouldn’t be considered as long as there was an ULO on board.
I have read what the others here have offered. What isn’t mentioned is that the Navy has small ships and boats, and non-combatant ships. While the chain of command– as it is often referred–doesn’t change, the commander of the ship may be actually a Lieutenant Commander, or even a Lieutenant, for a ship.
Let’s take for example: A destroyer escort may have a Lt. Commander as captain. While typically, a Commander or maybe Captain may be the captain, a Lt. Commander may be an interim captain–for some reason,–or the Lt. Commander may be the captain for older, smaller ship.
As for something like a seagoing Patrol Boat, a Lieutenant, Lieutenant, Jr. grade, or less likely an Ensign may be in charge. For a River Patrol Boat, it isn’t unusual to have a Chief Petty Officer, or Warrant Officer in charge of the boat. The same is true for a tug boat, whether seagoing or channel tug boat, usually the commander isn’t a true Captain, but certainly could be, for some reason.
However, these boats have a place they call home, and a Captain may be the Commanding Officer of the facility or the station.
As for supply and repair ships, there is a reduced line officer command, since it doesn’t have as much to do with combat, and more to do with running the ship. It probably has a Captain as Commanding Officer, though.
Submarines of old used to have Lt. Commanders or Commander running the show, but since the new fleet is large and the “boats” are big, a Captain is probably in charge. This last part I am speculating since my knowledge of submarines hasn’t been since the early 1970’s.
The whole point is that it is possible in a tragic event, in short order, a senior non-commissioned officer could be in charge!
The chain would be determined by seniority of the surface line officers, that is, the officers who are eligible for command. This excludes the staff corps and restricted line officers, such as supply officers, medical officers, or engineering duty officers. On a destroyer, cruiser, or frigate, the only staff corps would be the Supply Officer and the Disbursing Officer. The Disbursing Officer is a junior officer in the Supply Department who deals with the ship’s finances.
So if the Captain is incapacitated, the next in line would be the Executive Officer, followed by the department heads, in order of seniority, then the division officers, in order of seniority. Seniority is determined by year of commissioning. For officers within the same year group, seniority is determined by lineal number, which is a ranking of the line officers Navy-wide.
So if the Captain is incapacitated and there is an Admiral aboard who is staff corps, such as a Medical Officer, command would still go to the Executive Officer, who would be junior to the Admiral.
A Mustang, or Limited Duty Officer, is a former enlisted member who is directly commissioned to work in the area of his enlisted specialty. He has the same rank structure as line officers, but is not eligible for command. Warrant Officers are similar to Limited Duty Officers, but have a different rank structure.
In the case of a catstrophic event in which all line officers are incapacitated, I guess it would go strictly by seniority, including restricted line officers and enlisted.