We present Monte Carlo simulations of the isotropic-polar (IP) phase transition in an amphiphilic fluid carried out in the isothermal-isobaric ensemble. Our model consists of Lennard-Jones spheres where the attractive part of the potential is modified by an orientation-dependent function. This function gives rise to an angle dependence of the intermolecular attractions corresponding to that characteristic of point dipoles. Our data show a substantial system-size dependence of the dipolar order parameter. We analyze the system-size dependence in terms of the order-parameter distribution and a cumulant involving its first and second moments. The order parameter, its distribution, and susceptibility observe the scaling behavior characteristic of the 3D Ising universality class. Because of this scaling behavior and because all cumulants have a common intersection irrespective of system size we conclude that the IP phase transition is continuous. Considering pressures 1.3 ≤ P ≤ 3.0 we demonstrate that a line of continuous phase transitions exists which is analogous to the Curie line in systems exhibiting a ferroelectric transition. Our results are qualitatively consistent with Landau's theory of continuous phase transitions.