The single-particle density of states (DOS) is investigated for a system of localized electrons with Coulomb interaction in a random potential situated on a fractal lattice of the Vicsek type with a fractal dimension of two embedded in three-dimensional Euclidean space. To check the universality hypothesis of Efros we calculated the DOS numerically and compared it with the results for the square lattice model. We found that the DOS is determined by the geometric fractal dimension of the lattice instead of the spectral dimension, which usually determines the DOS. In particular, we found that the DOSs of the fractal lattice model and of the square lattice model show the same behaviour within the Coulomb gap provided that the ratio between the strengths of the random potential and the Coulomb interaction is identical for the two models. Thus the universality hypothesis of Efros is found to be valid with respect to different lattice structures.