When referring to the RAF most usually bring the Battle of Britain into the argument, quite rightly too. But it is worth remembering that SINCE WW2, all British Military air to air victories have been achieved by Fleet Air Arm aircraft flying from aircraft carriers. The RAF hasn't shot anything down since the forties! Also remember the RAF didn't show up until WW1 was nearly over (April Fools Day 1918), and was created by merging the Royal Naval Air Service (which had invented strategic Bombing) with the Royal Flying Corps. The impetus for this merger was the inability of both services to counter the German Zepplin raids, mainly because both services were preoccupied with the western front. So home air defence was the original mission of the RAF, not Strategic Bombing. All the Navy airmen who were behind Strategic bombing were absorbed into the RAF along with those who appreciated the need for shipboard aviation, which is why the Navy was short of 'air minded' officers between the wars.
Putting Naval Aviation in the hands of a land based Air Force with other priorities has been done three times in the last ninety years, always disastrously. The Navy got the Fleet Air Arm back in the late thirties, but the years under the RAF meant Carrier aircraft had lagged behind other nations naval air arms. The Japanese entered WW2 with the excellent Mitsubishi Zero, we had the Swordfish biplane! By the 60s we had the second most powerful Navy in the world with five strike carriers and two commando carriers, then an idiot in a government of idiots cancelled them and said the RAF could do the job cheaper. They couldn't do it at all, so the Invincible class and the Sea Harrier had to be bouht to retain air cover over the fleet.
In the last few years the Navy was forced to give up it's Sea Harriers because the RAF was having budget problems, leading to Joint Force Harrier, supposedly a 50/50 split of RAF/FAA manning. All sqns were supposed to rotate through Land ops and Carrier ops, but noticeably for the last four years our carrier decks have rarely seen any jets (unless they were USMC Harriers, Spanish Harriers or Italian Harriers!) This seems to be changing for the better now since the RAF's Tornado force has been sent out to the 'Stan at long last. In general if pilots and ground crew want to go to sea they join the FAA, those that don't gravitate to the RAF. That's a godd enough reason for the two air arms to exist separately. In the past, RAF aircrew who wanted to serve at sea could volunteer for a tour with the FAA, now they don't get a choice about sea service, they just grin and bear it. This leads to retention problems, and good aircrew are always in short supply. Also at this point, cutting the Carrier program won't save a penny, most of the money has been spent already and the penalty clauses against cancellation will mean it's actually cheaper to just build them! Without the Carriers, there is no Navy, the 1st Sea Lord understands this only too well and has fought tooth and nail to keep them. Cancellation will also mean an end to British Warship Building capability, as there is nothing else that can be ordered in their place to keep the yards open. Putting tens of thouseands of skilled shipyard workers on the dole helps the economy how exactly? It just puts an extra burden on the welfare budget. So cancellation is a major COST not saving.
Someone earlier posted 'Why do we need a Navy'. Let me educate you. Why would an Island nation 90% of whose trade travels by sea (and not just across the channel either) to all points around the globe and is utterly dependent on that trade for it's very survival need a Navy? If you think a 'Coastal Waters' defence policy would be OK then you have already surrendered to anyone, be they nation state or terrorist group who chooses to attack our Sea Lanes Of Communication (SLOC) You don't need U-Boats to do this either, planting bombs on merchant ships and sinking them mid ocean isn't a big challenge. I know from personal experience that UK port security is a sad joke, and the only reason they haven't been attacked is that the terror groups are as 'Sea Blind' as most people in this Island Nation. We have to be able to defend our supply lines first and foremost or we are finished.
So, the basic problem is not 'should we abolish one of the services' but 'we should acknowledge the importance of all the services and fund them accordingly', a message the Treasury is terrified of the British Public recieving!