The Schulze-Hardy rule, a well-known relation on colloidal dispersions between the critical coagulation concentration (ccc) and ionic valency of added electrolytes, is shown to hold in monolayer systems. A carboxyazobenzene derivative with a relatively short alkyl chain forms only a kinetically stable monolayer at the air-water interface without developing measurable surface pressure. It features the novel band structure on mica as Langmuir-Blodgett films, 1.5 nn thick, 60 nm wide, and can be longer than 100 mum. Atomic force microscopy reveals that the bands coagulate critically when the ion concentration in the water phase reaches a critical value. For monovalent ions, Na+ and Li+ exhibit critical coagulation, while K+ and Cs+ develop destabilizing hydration forces on mica before critical points are reached. Both La3+ and Ce3+ cause critical coagulation at much lower concentrations. For divalent ions, ccc's are in the order Cd2+ << Mg2+ similar to Ba2+ < Ca2+ < Sr2+. By comparing the experimental adsorption isotherms with a Poisson-Boltzmann theory that assumes charge regulation by metal-carboxylate complex formation, Ba2+ is identified to interact anomalously among alkaline earth metals. For hard Lewis acids, it is empirically found that ccc is, related to the ionic radius by a power law relationship, which is used to take into account charge regulation on the film. Then, ccc is shown to be inversely proportional to the ninth power of the ratio of the ionic radius to valency.