No, hydrogen and oxygen will not combine to form water unless you add energy. And with enough energy, even basic elements can be changed through a process called transmutation.
As for your claim that the amount of water on this planet is constant, I suggest you read up on comets.
This is a little misleading, carmanintx. Combining hydrogen and oxygen to form water releases ernergy, once you get the reaction started. That's why you can run a car on a tank of hydrogen and dribble water out the exhaust.
Breaking water apart into its components consumes energy -- basically the same amount of energy that's released if you recombine them.
You've probably seen water electrolysed then recombined in elementry chemestry class.