Green’s Dictionary of Slang

(US) a Catholic priest.

1897		J. London ‘The Road’ in Hendricks & Shepherd Jack London Reports (1970) 311–21: Their argot is peculiar study. [...] galway – priest – from the Gaelic.
1910		J. Sullivan ‘Criminal Sl.’ in Amer. Law Rev. LII (1918) 891: A Catholic priest is called a ‘Galway’ or a ‘buck.’.
1921		P. & T. Casey Gay-cat 54: A Galway named Father John Bresnahan, he tole us we oughta put by some money for a rainy day.
1936		Mencken Amer. Lang. (4th edn) 582: A Catholic priest is a buck or Galway, and the Salvation Army is Sally Ann.
1949		Monteleone Criminal Sl. (rev. edn).