Not an historian, but as a boxing fan I've read embarrassing amounts about those movies. The movie was viewed poorly by the Soviet public, but to my knowledge it didn't cause much serious outcry - at best it was seen as a harmless B-movie, at worst, shameless propagandizing. The initial translation into Russian was somewhat poor, and this probably caused fewer people to take the film seriously.

The choice of very Aryan Dolph Lundgren to play Drago seems to have been more widely panned than the allegations of PED use. He looks like the tall Swede that he is, and the conflation of anti-Soviet and anachronistic anti-Nazi imagery in the film was not well-regarded.

Given the sheer ubiquity of PED use in combat sports, the implication that American athletes are completely clean and Soviet athletes are completely dirty is not even plausible. For an individual, steroid use is a serious accusation - but the depiction of state-sponsored PED use in Rocky IV is so comically black-and-white that it's hard not to view it as clumsy propaganda.

Rocky IV was never shown at soviet cinemas. At the same time there were articles about this movie in the soviet magazines and newspapers, most of them were very negative.

For soviet movie critics it was not a harmless B-movie. It was an american propaganda.

Anti-sovietism of Rocky IV was good enough, but Rocky was played by Sylvester Stallone who also played Rambo, and Rambo III (1988) was about Afghan war.

Just a few quotes from "Soviet screen" magazine (one of the main movie magazines at the time):

"Rocky IV is just like Rambo, it's a fruit of Sylvester Stallone' sick imagination." «Роки-IV» следует по пятам за «Рэмбо», плодом больного воображения исполнителя главной роли, режиссера и сценариста обоих фильмов Сильвестра Сталлона.

"Rambo and other similar films aren't just trying to replay old battles (for example, to "win" Vietnam war where Pentagon has covered itself with indelible disgrace). Rocky IV is calling for the new military ventures; Rocky IV ads are saying: get ready for the World War III" «Рэмбо» и ему подобные не просто пережевывают и переигрывают битвы прошлого, одерживая сегодня «победу» во Вьетнаме, где Пентагон покрыл себя в прошлом несмываемым позором. «Роки-IV» звучит призывом к новым военным авантюрам, и в рекламе фильма так и сказано: «Будьте готовы к третьей мировой войне».

Actually, the last trivia is true. The real tagline was "Get ready for the next world war".

So, the official point of view was very negative, and it was almost impossible to watch this movie in 1985.

But Rocky/Rambo franchises became very popular with 'videosalons' (1986-1992), and most movie-goers didn't care about political messages at all.

link to the full article from Soviet Screen, 1986 (it's in russian, sorry): http://propagandahistory.ru/2002/ZHurnal-Sovetskiy-ekran--o-filmakh-Rembo-i-Rokki/

Tagging on to your question, was the Russian release of Rocky IV altered in any way (was that scene even shown)?