The rigorous calculation of the spin-orbit terms in a three-quark system is realized based on the Gaussian expansion method and the infinitesimally-shifted Gaussian basis functions in the framework of the relativized quark model, by ignoring the mixing between different excited states. Then, the complete mass spectra of the singly heavy baryons are obtained rigorously, under the heavy-quark dominance mechanism. Accordingly, systematical analyses are performed for the reliability and predictive power of the model, the fine structure of the singly heavy baryon spectra, the assignments of the excited baryons, and some important topics about heavy baryon spectroscopy, such as the missing states, 'spin-orbit puzzle,' and clustering effect. The results confirm that, under the heavy-quark dominance mechanism, the relativized quark model can describe the excitation spectra and the fine structures of the singly heavy baryons correctly and precisely.