The astrophysical S factor for the radiative capture d(p,γ)^{3}He in the energy range of interest for big bang nucleosynthesis (BBN) is calculated using an ab initio approach. The nuclear Hamiltonian retains both two- and three-nucleon interactions-the Argonne v_{18} and the Urbana IX, respectively. Both one- and many-body contributions to the nuclear current operator are included. The former retain for the first time, besides the 1/m leading order contribution (m is the nucleon mass), also the next-to-leading order term, proportional to 1/m^{3}. The many-body currents are constructed in order to satisfy the current conservation relation with the adopted Hamiltonian model. The hyperspherical harmonics technique is applied to solve the A=3 bound and scattering states. Particular attention is paid in this second case in order to obtain, in the energy range of BBN, an uncertainty on the astrophysical S factor of the order or below ∼1%. Then, in this energy range, the S factor is found to be ∼10% larger than the currently adopted values. Part of this increase (1%-3%) is due to the 1/m^{3} one-body operator, while the remaining is due to the new more accurate scattering wave functions. We have studied the implication of this new determination for the d(p,γ)^{3}He S factor on the deuterium primordial abundance. We find that the predicted theoretical value for ^{2}H/H is in excellent agreement with its experimental determination, using the most recent determination of the baryon density of the Planck experiment, and with a standard number of relativistic degrees of freedom N_{eff}=3.046 during primordial nucleosynthesis. This calls for a more accurate measurement of the astrophysical S factor in order to confirm the present predictions.