We use the functional renormalization group method to study a three-orbital model for superconducting Sr_{2}RuO_{4}. Although the pairing symmetry is found to be a chiral p wave, the atomic spin-orbit coupling induces near nodes for quasiparticle excitations. Our theory explains a major experimental puzzle between a d-wavelike feature observed in thermal experiments and the chiral p-wave triplet pairing revealed in nuclear-magnetic resonance and the Kerr effect.