U.S. Navy Chief Petty Officers typically address each other by their first name, not by their rank due in large part to the bond they all share in the Chiefs Mess. It's highly unlikely that the Doc, who is also a Chief would have addressed Chief Bell by his rank. He would have addressed him by his first name.
It is unlikely that a ship in the Pacific Ocean would be in water only 180 feet deep - equaling 30 fathoms - unless it were close to shore. The Navy ship sends a radio message that its position is 09.3000° S, 160.4800° E. That would place it in about 5600 feet of water and 10 miles from land.
There is a belief that the rank of captain determines the rank of the medical officer aboard a destroyer. In fact, all destroyers up to an including the latest (2016) Arleigh Burke class destroyers are manned by enlisted hospital corpsman. In the case of the larger Burke class, there are two corpsmen. But the smaller classes that went before, Knox class, Fletcher class, Sherman Class, etc.. All had a single enlisted hospital corpsman aboard. These corps are well trained for six months prior to being deployed. This contrasts with much shorter Army medic training.
There is a belief that the U.S. Navy stopped using the Copper Hard Hat diving helmet by 1962. When in fact, the Mark V hard hat was used by the US Navy from 1916 till 1984. In 1984 The older copper hard hat was replaced by a fiberglass version that continues to this day. There are many jobs that Navy divers perform that a diver with SCUBA gear could not perform. The heavy diving apparatus allows a diver to walk and work in higher currents without the fatigue that affect a SCUBA diver.
"Doc" Matthews is an NCO corpsman. A ship large enough to be commanded by an officer of staff rank (like Beecham, who wears the insignia of a commander or lieutenant commander) would have a surgeon of commissioned officer rank.
The diver, while at the wreck of the submarine has nearly unlimited visibility. There are times (usually in very cold, arctic, and antarctic waters) where crystal clear visibility extends to these kinds of depths. But at this depth in the Pacific, it would be rare.
There is never any indication of movement out the porthole in sick bay. A ship in the open ocean would be riding the waves as they rose and fell. There is not the slightest indication of motion. Something that would be unlikely in the Pacific ocean.
The goof item below may give away important plot points.
The diver McClure finds some dog tags on the sub and brings them to the surface. He knows they belong to Chief Bell when he hands them to the Captain. However, the Captain has to rub off dirt/corrosion before he can read them. If they were too dirty to read, how was McClure able to know who they belonged to?