Theoretically, If it rained exactly one inch, many gallons of water would there be in a square mile?

"A little over 17 million (17,378,743) gallons of water falls per square mile if the water is one inch deep."

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It's easier if you convert to the metric system first.
1 Inch = 0.0254m
1 Sq mile = 2.59 sq km = 2 590 000 sq m
65786 m3 = 65 786 000 Litres

1 liter = 0.26417205 US gallons
1 liter = 0.21996919 UK (Imperial) gallons
Converting back
= 17 367 500 US Gallons
= 13 815 100 Imperial Gallons

Reference For converting liters into gallons
http://wiki.answers.com/Q/How_many_liters_are_in_one_gallon

This concept is similar to the "acre-foot" used in irrigation allocations, which is the volume of one foot (twelve inches) of water spread across an acre of land. One acre-foot equals 1233.4818375475 cubic metres, or 325851.42857143 US gallons, of water. Dividing these by twelve (for an "acre-inch") gives us 27154.285714286 US gallons, which is provided (along with many other conversions) at http://online.unitconverterpro.com/conversion-tables/convert-alpha/factors.php?cat=volume&unit=1&val=

A square mile holds exactly 640 acres, so 640 x 27,154.285714286 = 17,378,742.8571264 US gallons (or 14,470,830.538448 Imperial gallons, or 65,785.69800253 cubic metres).