The following information mostly from the extensive entry in  The Fiddler's Companion:

Although Irish in origin (supposedly c. 1770-1780), the tune seems to have first appeared in print in Scotland, in Aird's 1787 collection (as "Auld Bessy").  As Bruce mentioned above, it was used in a popular pantomime in 1800 and thereafter gained wide popularity throughout Britain and Ireland and, subsequently, America.

"In the United States it was adopted as a favorite marching air by General George Custer's 7th Cavalry, an association which helped to popularize the jig throughout [the] country following Custer's demise.  'It had been said that the 7th acquired the song through Captain Miles Keogh, an Irishman and a former member of the Papal Guard, but it seems unlikely that (its American use) can be ascribed to a particular person, since 'Garryowen' appeared in a number of Civil War songsters, and was therefore presumably well known to any number of American soldiers in 1861-1865 -- dates preceding Keogh's association with the 7th" '  (Winstock, 1970; pgs. 102-104).

The tune is also used for a North West Morris dance.  There are a number of broadside copies of at least 3 songs set to the tune at the  Bodleian Library Broadside Collection.

Malcolm