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.
"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 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?