Capillary networks are essential in vertebrates to supply tissues with nutrients. Experiments of in vitro capillary formation show that cells randomly spread on a gel matrix autonomously organize to form vascular networks. Cells form disconnected networks at low densities and connected ones above a critical density. Above the critical density the network is characterized by a typical mesh size approximately 200 microm, which is approximately constant on a wide range of density values. In this paper we present a full characterization of a recently proposed model which reproduces the main features of the biological system, focusing on its dynamical properties, on the fractal properties of patterns, and on the percolative phase transition. We discuss the relevance of the model in relation with some experiments in living beings and proposed diagnostic methods based on the measurement of the fractal dimension of vascular networks.