Here is a post by Bruce Olson to Ballad-L about "Rosin the Beau".

  Note that an early sheet music issue of "Rosin then Beau" (not the two
  1838 ones) in the Lester Levy Sheet Music Collection (on their website)
  calls the song 'A Southern Ballad'.
  
  On the tune: There's basically nothing new below here except figuring
  out what tunes some of Bayard's sometimes awkward references
  actually refer to.
  
  S. P. Bayard, 'Dance to the Fiddle, March to the Fife', #620,
  traces the tune to a "Gigg" following "The Lowlands
  of Holand" in James Oswald's 2nd volume of 'A Collection of
  Curious Scots Tunes', n.d. [1742]. I have not found a copy of
  that work, but it's repeated (again perceeded by "The Lowlands of
  Holand") in his 'Caledonian Pocket Companion', II, c 1745. Bayard
  suggested that the tune was composed by Oswald himself (as for
  other 'Gigg' and 'Gigga' Oswald scattered around in these works), and
  I consider that quite likely also. While I had copied "The
  Lowland of Holand" from CPC, I neglected to copy the 'Gigg', and
  I was remiss in failing to note down my source for the ABC below,
  but am reasonably certain that it was supplied by Jack Campin,
  after my request for it on the Scots-L newsgroup.
  
  X:1
  T:"Gigg" (next to last tune, following "The Low Lands of Holand")
  S:CPC book II p36
  Q:1/4=120
  L:1/8
  M:6/8
  K:A mixolydian
  E| A2 A  (Bc)e| f3    a3 |(ec)A (AB)c|(Bc)A (B^G)
  E|(AB)A  (Bc)e| f3    a2f|(ec)A TB2 A| A3-   A2:|
  e| a2 a (^ga)g|Tf3 {f}e2c| B2B  (Bc)e| a2^g Tf3 |
     a3    (ec)A|(Bc)e  f2a| efA  TB2 A| A3-   A2:|]
  
  Bayard also found the tune in the 2nd strain of "Dumfries House"
  in vol. 1 of Gow's 'Complete Repository' (and this would have
  been known to many fiddlers). After the 1st (1799)
  edition this was attributed to John Riddle.
  
  X:2
  T:Dumfries House [by Jo. Riddle - in 3rd ed]
  S:Gow's 'Complete Repository, vol. 1 [1799 for 1st ed]
  S:2nd strain only here
  Q:1/4=120
  L:1/8
  M:6/8
  K:G
  B/c/|dBd dBd|gag "tr"f2e|dBd d3/2c/2B|AFDD2B/c/|dBd bBd|\
  gag"tr"f2e|dcB AGF|"tr"G3/2G2B/c/|dBd dBd|gag"tr"f2e|\
  dBd dcB|AFDD2B/c/|dBd ece|fdf gfe|dcB Agf|gdB G||

(Yes that was me, we'd exchanged emails about it).

So: American adaptation of a Scottish tune. Ireland doesn't come into it.