Interesting piece in Wiki on this. Contains the paragraph

The music used by Lawrence was borrowed from a Ukrainian folk-song "Oi ne khody, Hrytsju",[2] which is in turn based on a melody by Catterino Cavos from his vaudeville The Cossack-Poet.;[3][4][5][6] In its first appearance in the Cavos vaudeville the melody had an entirely different text: "Yes, of course, he is my lover..." ("???, ???????, ?? ??? ?????...").[7] Lawrence's parents were Ukrainian-Jewish immigrants who had immigrated to the United States from Bila Tserkva, Kyiv oblast, Ukraine.

I have understood it to be of Ukrainian origin for a long time and was looking up that folk song for a friend, I found this thread but lots of the links are now not avaialable so I did my own seach and found

Oi Ne Khody Hrytsiu - Theresa Sokyrka and Zirka

I love this version although it may not be to everyone's tastes :-)