Not really (although you could argue that common usage and ubiquity makes it essentially fine to get away with).

The USSR, or Union of Soviet Socialist Republics, was founded in 1922, combining several states which existed under the control of the Communists as a result of the Russian Civil War. The largest of these was Russia SFSR, but also included were Ukrainian SSR, Beylorussian SSR, and Transcaucasian SFSR (which would be broken up into several Republics later on). The USSR, quite literally, references the Uniting of these constituent Republics into a Federation. More would be added down the line, and by the time the Soviet Union began to crumble, there were fifteen of them. So essentially, during the period of the USSR's existence, Russians were Soviets (well, I know some ex-pats who would disagree with that, but I believe you understand my meaning), but Soviets were not necessarily Russian. And as such, "Russian" was not the proper term to refer to the Soviet Union as a whole, just Russia SFSR.

Edit: A reasonable comparison I think would be the penchant for people to say "England/English" when they mean "Britain/British/United Kingdom".

Russia was one country in the Soviet Union. It was the most important, most powerful, most prominent country, and it was where the seat of the Soviet power system was. It was where Soviet power started (in 1917), and spread from (in 1922 and onward). In some contexts you can use the terms interchangeably, because they overlap enough. In many you cannot.

Stalin, for example, was not Russian — he was Georgian in origin. Lysenko was Ukrainian in origin. They are both Soviet, but calling them Russian would be incorrect. Talking about "Russian nukes" in that period might make one not realize that the nukes in question, while developed in facilities wholly located in Russia, were stationed also in Belarus, Ukraine, and Kazakhstan (and much of the testing took place in Kazakhstan). Chernobyl was not a Russian reactor — it was in Ukraine, and that has important post-Soviet consequences.

I have a friend who is a Russianist who gets very irritated with people when they use "Russia" for "Soviet" interchangeably, because it under-emphasizes the divisions that existed, some of which were quite important for Soviet politics and history, and all of which are very important for post-Soviet politics and history, when those individual republics became independent countries once again.

This can look awfully pedantic to the man on the street, but for precise, learned conversation, it is often an important distinction worth getting right.