If I make a (19 X 19) board with 1" X 1.07" spacing between lines, how large should my stones be?
Depends on what you like. Current fashion is to have the stones so large that they barely fit and touch their neighbours, so if you want that, make them about 1 inch. Older boards had stones much smaller. In the end, the question is an aesthetic one, so do what you find most visually pleasing.
Are you planning to make your stones? If you are buying them there are basically two options, Chinese size and Japanese size. Japanese stones are about 22mm wide, Chinese about 23mm. Chinese stones will not fit well on a Japanese dimensioned board, too much overlap of stones. Your 1:1.07 ratio of width to length of the cells is close to Japanse style. Here's a link to a (definitive?) essay on Japanese board dimensions: http://www.gungfu.de/go/gobans/texte/Goban%20dimensions,%20a%20summary