I don't know about the first public toilet, but a public toilet was the first service action by a small group of friends in Chicago, who formed themselves into a social club.
Back in 1905, young lawyer Paul Harris was unmarried, lonely, and living in an unfamiliar place -- having grown up in a small town in Vermont. He invited three friends to his office for a social evening. The experience was so pleasant that they resolved to repeat it weekly, rotating the meeting place among their offices. By the second or third meeting, they decided that their affiliation should have a purpose greater than companionship. They decided to perform various services to the community, making the city a better place for all.
Their first project was to erect a public toilet on a downtown street, which was a much needed and greatly welcomed contribution. they went on from there, increasing the size of the group, doing more services, and soon expanded into an association of similar groups which they founded in other cities.
Theirs was the first Rotary Club. The organization grew into Rotary International -- now with nearly 33,000 clubs in over 160 countries, with one and a quarter million members. Their example led to formation of similar service clubs, which became such groups as Lions, Kiwanis, Optimists, and many more.
From that first public toilet, Rotary has provided uncounted services to communities, to vocations, and to the needy around the world. Rotary has by far the world's largest private scholarships program. Rotary initiated a program to rid the world of polio, and has immunized well over two billion children -- so that polio now survives in only three or four countries, and is rapidly moving toward complete eradication. Rotary clubs have provided countless water wells, new homes, schools, clinics, emergency vehicles, and on and on; and have provided vital relief in thousands of earthquakes, floods, tsunamis, hurricanes, fires, and many other disasters.
All because four guys decided to provide a public toilet.