No, people on ships (at least during the period I study) did not generally rely on fish as a main source of food -- at least not fresh fish, unless they could fish or buy fish inshore. The reason is that most of the world's fisheries are in shallow waters near shore, or offshore in shallow waters like the Grand Banks where cold and warm water mix and lead to an explosion of plankton and other aquatic life. (I'm excluding modern trawler fishing here, as that source of deep-sea fish would have been out of range of contemporary ships).

The French and especially Spanish navies did use fish as a source of food, but mostly in the form of dried stockfish. Though no navy would refuse a bounty of flying fish or something caught when close inshore, no navy would rely on attempting to fish as a main source of food.

While I can't speak to the tall ship era, fishing was certainly used, both as entertainment and to supplement rations, by Royal Navy vessels in WWI. Several memoirs mention it. For example, the memoir of Thomas Dixon, surgeon of HMS Kent, describes catching a shark while chasing the German cruiser Dresden in March 1915:

When I return just catch, a young seven-foot shark going for the bait. Hook him firmly and then begins intense excitement. Have to play shark tenderly as I only have the hook on three strands of fine wire... Meanwhile about a hundred people are crowded round shouting out advice and trying to take the line... Gradually I worked him forward towards the cutter alongside which is scraping the ship's side. Then a rope with a noose is passed down by line and the cutter's crew put it over his head. ... Finally, the noose is drawn tight and he is hoisted onto the quarter-deck, where he kicks and turns and snaps... Arrange with cook for shark-fin soup tonight.

Dixon was a remarkably prolific fisherman, along with the rest of Kent's crew. They had previously tried to catch sharks on the 29th October 1914, rigging up a shark hook off Dakar, and would later do so off Socorro. On the 21st March 1915, Dixon joined a working party fishing from one of the ship's boats:

This afternoon went away in the cutter with a volunteer crew to fish close inshore. We intended to fish for the tiny fish we had seen yesterday. We soon started, with all sorts of odd ends of line and various sized hooks. At first Redhead with a tiny gut hook got all the luck with small 'perch' and 'roach'. The big hookers were, at first, jealous but suddenly a Scots fisherman from the Highlands pulled in a 10lb black fish and after that the big hookers carried the day. Every few seconds someone hauled a fish.

Other methods of fishing were used - on the 9th January, the crew of Kent attempted to catch fish using guncotton charges, but caught only a few fish. The mania for fishing was not confined to the Kent. Dixon mentions Captain Healey of the AMC Orama having 'fished for six months'. Nor was it confined to the Royal Navy:

No fish caught. The Japanese have blown up too many in the harbour. They exploded an old torpedo war-head once and got, amongst others, a rock cod of 120 lb.

While it might be expected that ships on foreign stations, away from bases where food could be loaded, would be the only ones to turn to fishing, it wasn't unknown for it to happen in the Grand Fleet, which spent the war in Scapa Flow or Rosyth. Dudley Pound, then Commander aboard the dreadnought St Vincent, mentioned fishing in his own diary:

We keep the sailors amused by putting up the seine net round the side and then they play football and cricket.

However, this seems to have been more a way to keep the crew entertained, rather than as a way to supplement rations.

Sources:

The Enemy Fought Splendidly: Being the 1914-1915 diary of the Battle of the Falklands & its aftermath by Surgeon T. B. Dixon R.N.V.R of HMS Kent., Thomas Dixon, Blandford Press, 1983

Dudley Pound in the Grand Fleet, 1914–15, Dudley Pound, Paul Halpern (ed), Naval Records Society, The Naval Miscellany, Volume VI, 2004