The stability of binary fluid mixtures, with respect to a demixing transition, is examined within the framework of the geometrical approximation of the direct correlation for hard nonspherical particles. In this theory, the direct correlation function is essentially written in terms of the geometrical properties of the individual molecules, and those of the overlap region between two different molecules, taken at fixed separation and orientations. Within the present theory, the demixing spinodal line in the (ρ1,ρ2) concentration plane is obtained analytically, and shown to be a quadratic function of the total packing fraction and the compositions. The theory is applied herein to binary mixtures of hard spherocylinders in the isotropic phase. Isotropic fluid-fluid demixing can be predicted for a large variety of sizes and aspect ratios, and the necessary condition for entropic demixing is a sufficiently large thickness difference between the two particles that belong to each of the fluids in the mixture. As the theory reduces exactly to the Percus-Yevick approximation for a hard sphere mixture, accordingly it will not predict fluid-fluid demixing for this particular case. Demixing is also forbidden in two other cases; for a mixture of spherocylinders and small spheres, and for mixtures of equally thin spherocylinders. The influence and competition of an ordering instability on the demixing is also examined. The ordering of a fluid will always be displaced toward higher packing fractions by the addition of a nonordering fluid, and in some cases the entropic demixing can dominate the entire fluid range. Although the present theory merges exactly with the correct Onsager limit, it is shown that, for intermediate cases, the results can be significantly different from predictions of Onsager type approaches. These discrepancies are analyzed in particular for the needle plus spherocylinder mixture. Finally, in view of the nature of the theory, it is conjectured that the predicted demixing densities values are rather upper bounds to what should be expected.