We investigate numerically the critical conditions for detachment of an isolated, wall-bound pendant emulsion droplet acted upon by surface tension and wall-normal buoyancy forces alone. To that end, we present a simple extension of a diffuse interface model for partially miscible binary mixtures that was previously employed for simulating several two-phase flow phenomena far and near the critical point ["Phase-Field Approach to Multiphase Flow Modeling," Milan J. Math. 79, 597 (2011)] to allow for static contact angles other than 90°. Initially, we show that a formulation of the Cahn boundary condition based on a linear interpolation of surface tensions between the wall and each phase at equilibrium can be readily implemented in our numerical procedures by introducing a contact line indicator function and changes to our numerical algorithm that include an iterative method of enforcing inhomogeneous boundary conditions within a semi-implicit temporal scheme for the Cahn-Hilliard equation. Subsequently, the classical formulation of the Cahn boundary condition as first proposed by Jacqmin ["Contact-line dynamics of a diffuse fluid interface," J. Fluid Mech. 402, 57 (2000)], which accommodates a cubic (Hermite) interpolation of surface tensions between the wall and each phase at equilibrium, has also been incorporated into our numerical code. We show that the resulting phase-field models can be successfully employed for simulating three-phase contact line problems in stable emulsions with nearly immiscible components. We also show the first numerical determination of critical Bond numbers as a function of static contact angle by phase-field simulation. In addition, results of 3D simulations are compared to critical Bond numbers from a static stability analysis based on a numerical integration of the Young-Laplace equation. We argue that the discrepancy between our numerically determined static contact angle dependence of the critical Bond number and its sharp-interface counterpart is mainly due to the inability of the sharp interface analysis to describe the necking regime of drop detachment, where a sharpening of concentration gradients in the necking region produces an effective increase in (dynamic) surface tension, ultimately leading to a reduced tendency to detachment or an increase in the critical Bond number.