I don't think that there is any proof that this custom originated in Ukraine, as it is too widespread all throughout Eastern Europe to have originated in a specific area.
I don't have anything other than informal observation to back this up, but I think that Augustin717 is right. Slavic culture, generally, attached a ritual importance to linen and towels. Towels are an important gift in Serbian culture, and folk songs of the area are full of references to the washing of linen. Towels (now sashes) are worn by the best man (kum) and stari svat at a Serbian wedding. It might be a pan-Slavic custom that then influenced neighboring peoples. I know of Catholic Hungarian homes where towels were hung on pictures of the Sacred Heart, etc.
I see, thank you. There appear to be several icons with these as part of the depiction. The Mystical Supper icon I have has a type of clothe draped over the church where the Christ and Apostles are sitting. Also, I have a few icons of St. Gabriel the ArchAngel in which he has what looks like a Jewish prayer shawl in his hair. I just wonder if these two depictions are related to the rushnyky? It does seem as though the covering of religious objects has it's beginnings in the Middle East.
The object you see in Gabriel's hair is a ribbon, which is also a standard element of the iconography of any Archangel. It's upturned ends signify the spiritual hearing of the angel, always attentive to the commands of God.
The cloth that you see in the icon of the Mystical Supper has a different origin. Such a motif has its origin in Roman decorative painting (do a Google image search on "Boscoreale" or "House of the Vettii" and you can see what I mean). Icon painting's "visual vocabulary" is most indebted to Late Antique style. Drapery hung in the opening of windows and doorways is one such motif (if you are interested in the development and use of this, look through some of the fresco images from Pompeii, and then do a search on the illustrations of the "Vergilius Vaticanus" and "Vergilius Romanus" manuscripts. Both are late Antique copies of Virgil's Aeneid. Do a search then for the mosaics of San Apollinare Nuovo in Ravenna, the consular Diptych of Flavius Felix, the procession of Empress Theodora in the Basillica of San Vitale, and finally on the iconography of the four evangelists as they are writing their gospel.)
That's not to say it might not have a more "spiritual" meaning. Depending on the prototype, one could also argue that it probably references the veil of the Jewish temple as well as the curtain in front of the altar.