Has the United States Air Force used Contra-rotating Propeller Aircraft?
In short, the US didn't need one. 

The main reason for the development of Tu-95 was the rather unique situation existing after WWII, when the Soviet Union needed a heavy bomber to carry the (then very heavy) nuclear weapons. Due to the lack of bases anywhere near contiguous United States, the Soviets were at a significant disadvantage when it came to nuking the US before the ICBMs came into picture (US had bases in Europe and in Pacific). At that time, heavy bombers were the only way to deliver a nuclear payload and it required the aircraft to be fast, have a intercontinental range and be able to carry a reasonable payload.

The first attempt in developing a long range bomber during the last days of WWII by the Soviets didn't work out and as a result, they simply reverse engineered USAAF B-29s which had 'landed' in Russia into the Tu-4 'Bull', which lacked the necessary range. In a way, the B-29 was the forefather of Tu-95. Further development continued on this airframe, culminating in the Tu-80 'Barge', which had the necessary range, but was considered unsuitable as it lacked speed, which would be fatal against jet interceptors.

The Tu-95 was developed at the same time of the B-52 'stratofortress'. Actually the Soviet Air Force preferred the jet powered Myasishchev M-4 and Tupolev had to convince Stalin about the Tu-95's suitability. During the initial design phase, the Tupolev bureau considered two versions- one with four turbojets and another one with turboprops. The turbojet version was discarded as it would've lacked range due to the fuel hungry turbojets (Tupolev turned out to be right about that as the M-4 was sidelined due to lack of range).

The payload and speed requirements required a very powerful engine (the Kuznetsov NK-12 is the most powerful turboshaft engine ever) and the contra-rotating props were selected to overcome the problems associated with the large diameter of the propellers.

Western countries also did a lot work with contra-rotating props, but the advent of jet engines and the complexites associated with the counter-rotating props meant few graduated from the prototype stage. The Huges XF-11, for example had counter rotating props initially; they were changed to conventional ones after Huges crashed in it. The Fairey Gannet and Avro Shackleton were among the few western contra-rotating aircraft to see regular service (both were used in ASw roles, which highlight the increased endurance offered by them). A number of contra-rotating aircraft were tested during the end of WWII, but the advent of jet engines meant that the required speed could be achieved in a far better way.