A version of the two-dimensional site-diluted spin-(1/2 Ising model is proposed as a microscopic interaction model which governs solidification and growth processes controlled by vacancy diffusion. The Ising Hamiltonian describes a solid-fluid phase transition and it permits a thermodynamic temperature to be defined. The dynamics of the model are taken to involve (i) solid-fluid conversion and (ii) diffusion of vacancies in the fluid phase. By means of Monte Carlo computer-simulation techniques the solidification and growth processes following rapid thermal quenches below the transition temperature are studied as functions of temperature, time, and concentration. At zero temperature and high dilution, the growing solid is found to have a fractal morphology and the effective fractal exponent D varies with concentration and ratio of time scales of the two dynamical processes. The mechanism responsible for forming the fractal solid is shown to be a buildup of a locally high vacancy concentration in the active growth zone. The growth-probability measure of the fractals is analyzed in terms of multifractality by calculating the f(±) spectrum. It is shown that the basic ideas of relating probability measures of static fractal objects to the growth-probability distribution during formation of the fractal apply to the present model. The f(±) spectrum is found to be in the universality class of diffusion-limited aggregation. At finite temperatures, the fractal solid domains become metastable and a crossover to compact equilibrium solidification is observed as a function of both temperature and time. At low temperatures in the metastable fractal growth regime, the time dependence of the particle content of the domains is found to obey the scaling law, N(t)1/4tD/(2d-D-1). At higher temperatures where the growth is stable and leads to compact domains, the time dependence of N(t) can be described by a simple hyperbolic function. The various results of the theoretical model study are related to fractal and compact growth patterns observed in experimental studies of impure lipid monolayer films at air-water interfaces.