In fact, the U.S. did do a lot of work on AA missile systems, chiefly the Nike program. This included the Nike Ajax, Nike Hercules, and Nike Zeus. The latter was expected to counter ICBM launches. The program was scrapped in 1965 when it was determined that Soviet ICBMs would ultimately overwhelm any defenses, and that the only real defense was the MAD (Mutual Assured Destruction) doctrine, the same strategy that was lampooned in Dr. Strangelove. Also, the experiment was costly, delivering less bang for the buck (pun intended) than the administration could stomach, and defense funds were prioritized elsewhere, including to the burgeoning war in Vietnam.
Here are a few excerpts from an interesting website that describes the Nike Missile System:
Nike, named for the mythical Greek goddess of victory, was the name given to a program which ultimately produced the world's first successful, widely-deployed, guided surface-to-air missile system. Planning for Nike was begun during the last months of the Second World War when the U.S. Army realized that conventional anti-aircraft artillery would not be able to provide an adequate defense against the fast, high-flying and maneuverable jet aircraft which were being introduced into service, particularly by the Germans.
The first successful test firing of a Nike missile occurred during 1951. This first Nike missile was later given the name Nike "Ajax". Nike Ajax was a slender, two-stage guided missile powered by a liquid-fueled motor utilizing a combination of inhibited red fuming nitric acid (IRFNA), unsymmetrical dimethyl hydrazine (UDMH) and JP-4 jet petroleum. The Ajax was blasted off of its launcher by means of a jettisonable solid fuel rocket booster which fired for about 3 seconds, accelerating the missile with a power of 25 times the force of gravity.
The shifting nature of the Soviet threat meant that the air defense role, for which Nike was originally intended, became relatively less critical as time passed. Defense dollars were needed for other projects (including the development of American ICBMs and potential missile defenses) and also to fund the rapidly growing war in Vietnam.
The picture above may also be found at the linked site.