Truth be known, the "F" in VF was the abbreviation of the word "Fighting," not "Fighter." Squadrons were known as "fighting squadrons" and in both print and conversation an individual squadron would be known as, e.g., "Fighting 3" or "Fighting 11". These designations came about with the USN system for ship nomenclature approved by the Secretary of the Navy on July 17, 1920, in General Order No 541, ". . . with a view to the classification of all naval vessel and small craft so as to indicate the type and class to which assigned, to distinguish between those available for general fleet action and those suitable only for subsidiary service, and to provide official identification numbers which are to be employed in official correspondence, for the marking of spare parts, etc." The order established type designations and class designations for then existent ship and aircraft types. To quote No 541:
"HEAVIER-THAN-AIR-AIRCRAFT
Type Designation: V
Class Designations:
Fighting plane: VF
Observation plane: VO
Scouting plane: VS
Patrol plane: VP
Torpedo and bombing plane: VT
Fleet plane: VG"
By the time WW2 came along, the VT as representing torpedo and bombing plane types had devolved to just torpedo planes and the bombing types were designated as VB. The business of adding an "ing" to a type designator lasted through the war years, eventually giving way to the more modern, e.g., "fighter".
Squadrons, then, operating from carriers in the war years were referred to as fighting, scouting, bombing, torpedo squadrons . . . and starting in January 1945, the new type carrier squadron "VBF" which was referred to, in a reversal of its designating letters, as "fighting-bombing." It is the mark of an old hand or someone very, very, familiar with the subject to see squadrons referred to in this manner. Squadrons operating from cruisers were designated VCS or "Cruiser Scouting." Battleships used VO, and here we start to get grey as the generic type of duty was VOS you see them referred to as both "Observation Scouting" or "Observing." In my experience, "Observing" by practitioners, "Observation Scouting" by non. Cruisers were, of course, part of the Scouting Force, so their planes were Cruiser Scouting and battleships were the Battle Force so their aircrafts' primary mission was Observing. Note, though that torpedo squadrons did not have the "ing" on the end of the designation description. With that one exception, the "ing" thing appears to be related to squadrons operating aircraft which could be operated from ships. Land-base squadrons were referred to as Patrol, or Utility, or Rescue, or Photo(graphic) and so on, without the "ing."
It gets worse, just to keep things confusing, there's the shorthand. A fighting squadron would also be referred to as a "FitRon" or, sometimes "FightRon" and likewise you see the like of BombRon (sometimes BomRon), ScoutRon, TorpRon, PatRon, PhotoRon, RescRon, and so on with the number tacked on, e.g., "FitRon 11," "FightRon 85," or "BombRon 20."
Anyway, in the vernacular, in the WW2 period, a navy fighter squadron would be referred to as "Fighting (number)," thus VF-24 would be "Fighting 24" or "FitRon 24". In print one might easily find the 24 written out, Twenty Four.
All that being said, in late 1944, the designations did officially change, dropping the "ing" practice and substituting "er" in its place, so VFs became fighter squadrons, VB became bomber squadrons and so on. For the VCS and VOS, the "ing" simply disappeared. The pre-war practice of the "ing" pretty much disappeared from official writings, but the spoken "ing" survived in conversation into the early 1950s - at least for fighters, bombing and torpedo squadrons went the way of VA, with "A" for "Attack." Continued usage, one might suppose, would be considered "salty."
By October 1947, 'twas all settled with the publication of NavAer 00-25Q-13 "Model Designation of Naval Aircraft." This document reads:
"HEAVIER-THAN-AIR (FIXED WING) V
Fighter (destroy enemy aircraft in the air) VF
Attack (destroy enemy surface or ground targets) VA
Patrol (search for enemy) VP*
Observation (observe and direct ship and shore gunfire) VO
Transport VR*
Utility VU
Training VT
Gliders VG
"HEAVIER-THAN-AIR (ROTARY WING) H
Search and rescue HH
Observation HO
Training HT
Transport HR
Utility HU
"PILOTLESS AIRCRAFT K
Aerial Target KD
"GUIDED MISSILES M
Air-to-Air AAM
Air-to-Surface ASM
Air-to-Underwater AUM
Surface-to-Air SAM
Surface-to-Surface SSM
Underwater-to-Air UAM
Underwater-to-Surface USM
Test Vehicle TV
"LIGHTER-THAN-AIR Z
Patrol and escort ZP
Search and rescue ZH
Training ZT
Utility ZU
"*Note: For administrative purposes Class VP and VR are further classified into four-engine landplane, two-engine landplane, four-engine seaplane and two-engine seaplane and are further identified by adding the letters (HL), (ML), (HS) and (MS) respectively to the basic class designation."
Of course some of these designations eventually went away - the Z's come to mind as well as sub-types for VPs and VRs -and others were added - VS came back into use and others, such as VAQ, VAW were created, but the framework remains pretty much the same.
As far as the "V" in CV is concerned, the USN is not exactly sure just how that came about, however, they lean toward the French word "volplane," the verb tense of which, "voler," means "to fly." Remember, this scheme of type designations came after the Great War and the USN had fairly substantial air operations in France and Britain. A check of my trusty WWI era English-French dictionary finds for the English word "fly" given in French as "V. 1. Voler: vo"lë' {in the air}. flying. I. n. Le vol: lə vol. II. a. Volant (w..): vo"lân'; volante (/.): vo"lânt' - flying" certainly having flying connotations. If you were to visit the USN Naval Historical Command web site, some diligent sniffing about would lead you to
http://www.history.navy.mil/avh-1910/APP16.PDF which has a nice little discussion on the matter. My equally trusty 1935 Naval Phraseology volume shows for "the flight" in French "le vol," in Italian "il volo," and, in Spanish, "el vuelo." All nice little "V's".
Of course - sly grin - I have my own theory. French for airplane is "avion" and since the A was already taken by the heavy cruiser's CA in the 1920 designation scheme, they simply took the next letter "V" for the carrier's CV.
Rich