A quantitative model of ion-stabilized gas bubbles is suggested. Charging the bubbles by the ions, which are capable of adsorption, and the screening by a cloud of counter-ions, which are less absorptable, is modeled. It is shown that, subject to the charge of bubble, two regimes of such screening can be realized. For low-charged bubbles, the screening is described in the framework of the known linearized Debye-Huckel approach, when the sign of the counter-ion cloud is preserved everywhere in the liquid, whereas at large charge this sign is changed at some distance from the bubble surface. This effect provides the mechanism for the emergence of two types of compound particles having the opposite polarity, which leads to the aggregation of such compound particles into fractal clusters. Based on experimental data, arguments in favor of the existence of the clusters composed of the ion-stabilized bubbles in aqueous electrolyte solutions are advanced. This paper provides theoretical grounds for the experimental results presented in the previous paper (part I) published in this journal.