To put it simply, while dive bombers can be effective and accurate, they also have a lot of vulnerabilities. Purpose-built dive bombers were generally slow, with poor manoeuvrability. The typical attack pattern required them to fly straight and level at a high altitude , making them easy targets for flak. They were also vulnerable after pulling out of their dives. The RAF and USAAF had tactical attack aircraft like the P-47, Typhoon and Mosquito which were capable of making accurate attacks, while retaining the advantage of speed, and reduced flak exposure through low level approaches. However, Spitfires and P-51s were used for dive bombing with minor modifications.

I'd also like to correct you on the RN's use of dive bombers. They entered the war using the Blackburn Skua as a dive bomber and fighter. This aircraft became the first to sink a major naval vessel in a combat situation, sinking the German cruiser Konigsberg in Bergen harbour in April 1940. However, it was slow, underpowered, had poor manoeuvrability and was easily damaged. It was quickly retired. The Fleet Air Arm would never operate another aircraft specifically as a dive bomber. Their torpedo aircraft could dive bomb, but were generally operated solely as torpedo bombers. The Swordfish and Albacore could dive bomb with no need for dive brakes, while the Barracuda was specifically designed as a torpedo aircraft with dive bombing capabilities. The Barracuda was used as a dive bomber against the Tirpitz due to the difficulty of making torpedo attacks in the Kaafjord where she was moored. However, FAA doctrine saw the torpedo as the prime armament of its aircraft, and they were used as such. The FAA also operated the Corsair, which could be used for dive bombing, but was mostly used as a fighter.

Sources:

The Fleet Air Arm Handbook 1939-1945, David Wragg, 2003, Sutton Publishing.

George C Marshall, the Chief of Staff of the US Army was very impressed with how the Germans used Stuka dive-bombers during their 1940 campaign against France. He forced the US Army Air Force to buy Douglas Dauntless (A-24), Curtis Helldiver (A-25) Vultee Vengence (A-35) dive bombers in fairly large quantities. However, for the reasons /u/thefourthmaninaboat describes, the US Army Air Force did not like these single engine dive bombers. They did use the A-24 in the dark days of early 1942, in the Pacific theater, but the USAAF had their own ideas on what made an effective ground attack plane. They preferred the Douglas A-20, a twin engine war plane that could carry 4,000 pounds of bombs, strafe the target with four, fifty caliber machine guns and fly nearly 100 miles per hour faster than an A-24.
By 1944, the USAAF found the P-38 and P-47 to be very effective ground attack planes too. By then, the USAAF had developed bazooka rockets and napalm bombs that were far more dangerous to German ground troops than anything a dive bomber could deliver.

In Britain, naval aviation was under the control of the RAF during the inter-war years (the Fleet Air Arm (FAA) transferred back to the Admiralty in May 1939), and the RAF was generally committed to the idea of massed high-level attack by heavy bombers, against warships as well as land targets. The Naval Staff were keen to adopt a dive bomber in the 1930s, but FAA requirements got shunted to the back of the queue; combined with the limited space on carriers at the time, that led to the multi-role Blackburn Skua fighter/dive bomber, most involved well aware of the compromises necessary to fulfil both roles.

There was some effort to produce land-based tactical support aircraft, Air Ministry Specification P.4/34 resulting in the Hawker Henley in 1937, a light bomber with dive bombing capability, similar in design to the Hurricane, but with the pressing need for fighters for home defence, and general disinterest in tactical airpower within the RAF, the Henley was cancelled as a bomber, a small number being produced as target tugs.

As the war developed, as others have noted in comments the vulnerability of dive bombers to enemy aircraft and flak made them less attractive; even with the scarcity of Luftwaffe fighters in western Europe in 1944 the Germans fielded a lot of very effective light/medium 20mm and 37mm flak guns.

Somewhat ironically, the enthusiastic championing of dive bombing in Germany, principally by Ernst Udet, led to requirements for almost all their bombers to have dive bombing capability, including the He 177 heavy bomber that was entirely unsuitable for the role.

Thanks so much for the replies, /u/thefourthmaninaboat, /u/eidetic, /u/davratta, /u/KrazyEagle and /u/Bigglesworth_! As you can probably tell, I've read a lot on WWII in the Pacific, not as much in Europe.

Is there a good history of the air war in Europe, something like Bergerud's Fire in the Sky, that you'd recommend?

For the most part it was due to the vulnerability of dive bombers during their attack run, the effectiveness of anti aircraft guns, and a shift toward multi role aircraft. A dive bomber was never very maneuverable in the first place and might have been able to barely hit 300 mph unladen but with a bomb load both maneuverability and speed decreased, this combined with their attack pattern, flying straight and level at around 10000 feet to the target then diving, made the very vulnerable to anti aircraft guns and defending fighters during both approach and exit from combat. In the Pacific the Allies were able to use dive bombers very effectively against the Japanese because the Japanese were never able to mount effective anti aircraft guns on their ships, most Allied combat air losses in the Pacific were due to Japanese aircraft. But in Europe the Germans had much better anti aircraft guns and control of those guns, including radar control which the Japanese never mastered during the war, which would be deadly to aircraft even without the proximity fuse the Germans lacked. The effectiveness German anti aircraft guns combined with the vulnerability of dive bombers during their attack run ment few were ever deployed to Europe. All this combined with the appearance of multi role aircraft that could be dive bombers, torpedo bombers, ground attack aircraft, fighters, or interceptors as needed caused an elimination of several types of specialized aircraft.

From a recent question about carrier air wing tactics which was mostly about torpedo bomber tactics but touched on dive bombers gave a great link (I highly recomend reading the whole article it provides a good overview of dive bombers) that explained the end of the dive bomber. The first bolded paragraph touches on anti aircraft guns and the second on multi role aircraft.

THE END OF A WEAPON It was improvements in anti-aircraft fire that spelled the end of the dive bomber, however.

By the end of the war, it was practically suicidal for any dive bomber to approach an American ship. More (and better) AA guns, under radar direction, using proximity fuzes, could blow any plane out of the air at long ranges. It was bad enough for a torpedo plane, having to fly in a straight line at a relatively low speed, but at least they could stay a good distance away from their target and still release their weapon. For a dive bomber, however, which basically had to overfly its target, they were sitting ducks. By the end of the war, a Japanese naval vessel, which never carried a particularly large number of AA guns even under the best of circumstances, could expect that it would have the number of AA guns doubled or tripled from what it had carried at the beginning of the war.

And though the Axis powers never developed the proximity fuze for their AA guns, when you're putting enough bullets in the air you often didn't need them. German anti-aircraft guns were fearsome indeed, which is one of the reasons the Allies didn't use dive bombers much in Europe. Indeed, the RAF and FAA, who are often credited with creating and formalizing the concept of dive bombing, never had more than a couple of squadrons of dive bombers at any time during WWII. This despite the benefits of the type being on display against them in Poland, Belgium and France. The reason given was that it would be too costly in men and machines, and it's hard to debate the point.

Another reason for the disappearance of the dive bomber was the rise of what is now called the multi-role aircraft. The F4U Corsair, for example, could perform steep diving attacks nearly as well as the Dauntless despite the lack of dive brakes, had a longer range, and could carry a greater weight of bombs to boot. The P-47 Thunderbolt was another excellent fighter that could double as an attack plane, as was the Focke-Wulf Fw190. Using a fighter to do a bomber's job, and perhaps do it better to boot, made a lot of sense, particularly in the crowded hangar decks of an aircraft carrier.

The massacre of a squadron of dive bombers by such technology as a surface-to-air missile could easily be imagined, but by that time the dive bomber was no more.