I wish I knew who started that rumour; he needs a good kicking. Every time someone even mentions Greensleeves, or innocently revives an old thread on the subject, someone invariably trots out that Henry VIII business, for which there is not an atom of evidence, though it has been repeated so often as to have attained the status of unchallengeable received wisdom.

It is a myth. See RE: GREENSLEEVES ... Whence the name?, where Chappell's suggestion is dealt with. The thread otherwise covers quite some ground, and includes, besides a few contributions from people who have bothered to think before posting, a great deal of nonsense, including the frankly bizarre suggestion that the name Greensleeves has some relation or other to the Gaelic word sliabh. It has the advantage of novelty, but is about as likely as the whole song being about Belgian cheese.