Alan Lomax says definitely (without evidence) that:

"This refrain [to "Green Grows the Laurel"] was so much sung by the Irish-American troops who marched into Mexico during the 1846-8 war that Mexicans thenceforth called Americans 'gringos' (from 'green grows')." (The Folk Songs of North America, p. 326)

This may have consolidated the popular notion of the word origin.

~Masato