Yes, public water is possible within international trade agreements, because the world is large. There are not that many ships on the water compared to how vast the land is. Near a country, it is important that the country or less own the water, for national security purposes, but the rest of the water is enough to share.
Water is a basic human right that should not be controlled by private corporations. Without water, people can't survive. Government should be responsible for passing legislation that guarantees us to water, just like we are guaranteed freedom of speech, religion and the right to bear arms. It is best to have water as a public good and not bound to market forces that will drive the price up when demand is high and keep people of lesser means from having access to it.