Greensleeves persists in tradition as a Morris tune, usually in modal form. Kick My Arse is one name under which it has been found; as such, Cecil Sharp noted it in Wyresdale, Lancashire. Here is an .abc from Richard Robinson's Tunebook:
T:Kick my arse
N:The tune for the Wyresdale "Old man's dance" collected by Cecil Sharp.
c2c cde | d2B G2B | c2A ABc | B2G E2B | \
c2c cde | d2B G2B | cBA BAG | A3 A3 :|
|:g2g gfe | d2B G2f | g2g gfe | a2f d2f | \
g2g gfe | d2B G2B | cBA BAG | A3 A3 :|
As to the possible existence of the tune prior to 1580, Claude M. Simpson (The British Broadside Ballad and Its Music, 1966) had this to say:
"...The Lord of Lorne and the False Steward, licensed on October 6, 1580, is to be sung to Greensleeves or Greensleeves and Pudding-pies in seventeenth-century issues, the earliest that have survived... Chappell (Popular Music of the Olden Time I, 228) suggests that the ballad and hence the tune may be older, quoting the Satyra Prima of Edward Guilpin's Skialetheia, or a Shadowe of Truth, 1598:Yet, like th'olde Ballad of the Lord of Lorne,But though the ballad were familiar in the time of Henry VIII, we may not conclude that the tune Greensleeves is of equal antiquity, for we cannot be sure that The Lord of Lorne was originally sung to the tune. What we do know is that editions of the second half of the seventeenth century call for the tune. An earlier version in the Percy Folio MS is without tune direction."
Whose last line in King Harries days was born...
As has already been mentioned, Greensleeves was the big hit of 1580, rapidly spawning a whole series of spin-offs; Shakespeare mentions it in two of his plays. By the end of the 16th century, the term was "a metaphor for a handsomely dressed woman, or more usually a courtesan". Musicians from a number of countries enjoyed brief vogues at the Elizabethan court; there is no evidence of any Irish connection in this case. It is probably also unwise to try to read too much into what is not, after all, a very complex song.
The full text, and further information, can be seen in the Greensleeves History of thread.