I don't want to imply any influence here, but I do want to note that the octatonic scale is a symmetrical way of dividing up the octave, as opposed to the major and minor scales of common practice harmony, which are asymmetric ways of dividing up the octave. Composers like Bela Bartók were also exploring symmetrical alternatives, in his case it was often the tritone.
Unsurprisingly, I am not the first to notice this. Taruskin offers extensive quotes from Russian modernist critic Vyacheslav Karatïgin on the Rite of Spring and mentions that:
Like Myaskovky, Karatïgin professes to see a deep kinship between Stravinsky and Schoenberg--prompted in his case, no doubt, by the enthusiastic letter he had received from the Russian composer about the German. [Tarusin, Stravinsky and the Russian Traditions, p. 1029]Stravinsky heard Schoenberg's Pierrot Lunaire in Berlin on 8 December, 1912 and that was the occasion for the letter to Karatïgin where he says:
I saw from your lines that you truly love and understand Schoenberg--that truly remarkable artist of our time. Therefore I thought you might be interested to learn about his very latest composition, in which the whole extraordinary essence of his creative genius is most intensely concentrated [i. e. Pierrot Lunaire]. [quoted in Taruskin, op. cit. p. 824]Another commentator on connections between Schoenberg and Stravinsky was theorist Allen Forte who said in his 1978 study of The Rite that:
...in The Rite of Spring Stravinsky employed extensively for the first time the new harmonies that first emerged in the works of Schoenberg and Webern around 1907-08 [quoted in Taruskin, op. cit. p. 1022]In later years both Schoenberg and Stravinsky said a lot of critical, even derogatory things, about one another, which illustrates more the fact that they were career rivals.
For an envoi let's hear some of Pierrot Lunaire, one of the most unusual pieces in Schoenberg's output: