Margaret Bradford Boni's Fireside Book of Folk Songs (Simon and Schuster, 1947) may have helped to spread the etymology:

"An old Irish song, widely popular with the early Texas cowboys. A colorful fable holds that the Mexican word 'gringo,' meaning cowboy, was derived from the song, for the Mexicans referred to the Americans by the first two words of the title 'Green Grow,' pronouncing it 'Gringo.'" (p. 174)

Denis McLoughlin, in The Encyclopedia of the Old West (Routledge & Kegan Paul, 1977), says:

"GRINGO. Contemptuous Mexican name for Anglo Americans. The fact that 'Green Grow the Rushes' was a favorite nineteenth-century American song has given rise to the belief that 'gringo' is a more or less phonetic rendering of 'green grow'; this is an ingenious surmise and nothing more."(p. 204)

According to Peter Watts' A Dictionary of the Old West (Alfred A. Knopf, 1977, p. 150),

gringo
(from Sp griego: Greek, a term used for foreigners in past times). In the Southwest and Mexico, it was applied derogatively to Anglos, but one suspects it was employed at times to label any strangers other than Indians. Barrèr and Leland 1897 states that it was applied to a newcomer, a johnny-come-lately. (Davis 1857; Weseen 1934)

~Masato