Timing-wise, the US nuclear arsenal was very small in the years between 1945 and 1949, when the Soviet Union became a nuclear power. There are organizational and technical reasons for this, but in 1948 the US "only" had 50 nukes and in 1949 it "only" had 170. That sounds like a lot, but the US war planners had calculated it would require on the order of 120-470 bombs to fully wipe out the USSR with them. Anything less than that was going to be a very long slog of a war mixed with occasional nuclear detonations.

The reason they had so few was that the atomic complex developed in WWII was allowed to fall into disrepair in the immediate postwar due to Congressional bickering about the nature of postwar domestic control of the atom, plus technical problems with plutonium production (unanticipatedly large graphite expansion in the reactors) and lack of an accelerated development program (e.g. they did not test a composite Pu-U235 bomb until 1948, when they actually had them ready to test in 1946).

These early weapons were also quite crude — literally the same model as the bomb dropped on Nagasaki. (The Mark 4 bomb did not begin deployment until 1949, and it was "just" an upgraded version of the Nagasaki bomb, the Mark 3.) They were being fielded in B-29s, which have some ability to penetrate the USSR but primarily from bases that the US didn't really have control over. Anyway, it would not have been easy.

The US ability to really nuke the Soviets out of the water came in the 1950s, when it encircled the USSR with nuclear-armed bomber bases, longer-range bombers, and improved weapons. The Soviets had a nuclear capability by this time, but it was really only a deterrent against European targets — the Soviets had barely any ability to hit the continental US until the early 1960s when their ICBMs went online.