The file 'Tou lithou sphragisthentos' is a version of the Resurrectional Troparion Apolytikion that I have extracted from an old article by Dom M. Schwarz, 'Le Chant ecclesiatique byzantin de nos jours', Irenikon 10 (1933): 225-242, 335-352, 11 (1934) 168-195) (10: 337). Dom Schwarz transcribed the troparion from Nea Anastasimatarion (Bucharest, 1820). Transpose it up a third and remove the flat from the A and you will see what I mean.

The Greek melody alternates so-final and re-final phrases. One thing is lacking for a complete parallel to the Russian Greek Chant melody: there is not a tenor on fa in the second (re-final) phrase. Otherwise a simplification of the melody together with the tonal transformation would yield a creditable counterpart to the Russian melody. The opening figure, re-mi-fa, re-mi-fa-so-so-so, exactly parallels the familiar Russian do-re-me, do-re-mi-fa-fa-fa.

Now, to see that this is in reality an old form of the Greek melody, consult Christian Troelsgaard, 'The Repertories of Model Melodies (Automela) in Byzantine Musical Manuscripts', Cahiers de l'Institut du Moyen-Age grec et latin 71 (2000): 3-27. Example I, 1 (3) on pp. 9-10 consists of four versions of this melody. The fourth, from Anastasimatarion argon kai syntomon (Athens: Zoe, 1972), represents the version common nowadays; the resemblance to the Russian melody is much less evident, although re-final and so-final phrases are still present. The third version, from 1720, is much more like Dom Schwarz's example; here the phrases are written as la- and re- final and do not alternate as regularly as in the version of a century later, but it is clearly the same basic melody. And moving back in time to the second version (16th century) and even to the first (14th century) the continuity remains obvious.

If this were a unique case among the troparia, one might still dismiss it as coincidence; but in fact most of the troparia of the Grecheskii Rospev show essential resemblance to those of the Byzantine Chant. So Tchaikovsky was, IMO, really drawing indirectly from old Byzantine tradition.

I set the text of the file in SPIonic and the music in NoteWorthy Composer; I used PDF995 to distill the file.