If he married in US than the church marriage record should have a place of birth. The children church baptismal record as well should have his place of birth. I take it you have not located his arrival manifest?
Not many SWIATOWY arrived thru NY EI but worth your time to take a look at the manifests. Family went to family in the beginning so one of them could be going to the address of your ancestor.www.rootsweb.com/~polwgw/index.htmlwww.polishroots.comhttp://genforum.comhttp://www.moikrewni.pl/mapa/kompletny/%25C5%259Bwiatowa.htm...http://www.moikrewni.pl/mapa/kompletny/%25C5%259Bwiatowy.htm...http://www.moikrewni.pl/mapa/kompletny/swiatowy.htmlhttp://www.moikrewni.pl/mapa/kompletny/kazmierczak.html
map of Poland's wojewodztwos:http://www.geneteka.genealodzy.pl/