"Stabilized oxygen" contains nothing more than another kind of bleach. The makers of "stabilized oxygen" are just adding Chlorine Dioxide or Hydrogen Peroxide to water. So it's really the chlorine that is doing the disinfecting just as bleach or iodine would. There is some activity from the oxygen, but these are just bleaches. In some cases, perhaps, "stabilized oxygen" contains ozone, which is a disinfectant, but probably hard to get where you are and expensive.
Fundamentally, "stabilized oxygen" is not something very scientific but is in the realm of pseudoscience, or plain old marketing. Websites with information on "stabilized oxygen" refer to unaccredited universities with dubious researchers. The US Federal Trade Commission comments on "stabilized oxygen" on: http://www.quackwatch.com/04ConsumerEducation/News/vitamino.html.
If your goal is to purify water without using halides (Chlorine, Iodine, etc) I would suggest three alternatives. Boil your water, or use a microfiltration system, or distill it. There are camping stills and filters available commercially. The website http://www.netside.com/~lcoble/dir9/water.htm has suggestions.