I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For William McGarrity B 1835 +/- 4 yrs, Co. Galway, with the father's forename unknown, I found 1 rcd in 1838. Father's forename for this rcd was Alexander.
For William McGarrity M 1860 +/- 4 yrs in Co. Galway, I found 0 rcds. Expanding the search to all Ireland (in the database), I found 4 rcds: 1857 Westmeath, 1859 Mayo, 1859 Armagh, and 1863 Armagh.
Suggest you search church, immigration, naturalization, and census rcds to determine whether William married in Ireland or America. Also try to find placenames and/or parent's name(s).
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.