The question as posed above is incorrect. The critical density at critical temperature is NOT calculated on the basis of "scale factor universality". To calculate the critical density (corresponding to the liquid-gas transition), one has to calculate the Helmholtz free energy of the constituent atoms (for this, one will have to assume some form for the interaction potential) and solve the critical density from the conditions to hold at criticality. In the case at hand, expressing the density n in units of the critical density n_c and temperature T in units of the critical temperature T_c, the phase boundary, the coexistence curve, near criticality proves to be describable by a universal curve, independent of the details of the interaction. With t = T/T_c, the two branches n_+ and n_- of the coexistence curve (the former corresponding to n/n_c > 1 and the latter to n/n_c < 1) as measured from 1 are for t approaching 1 from below described by curves proportional to (1- t)^{beta}, where beta = 1/3. The actual value of n_c does not come into play.

For details, see, e.g., the excellent book by Chaikin and Lubensky (Principles of Condensed Matter Physics, Cambridge University Press, 2000).