We use a suite of semi-empirical models to predict the galaxy-galaxy merger rate and relative contributions to bulge growth as a function of mass (both halo and stellar), redshift, and mass ratio. The models use empirical constraints on the halo occupation distribution, evolved forward in time, to robustly identify where and when galaxy mergers occur. Together with the results of high-resolution merger simulations, this allows us to quantify the relative contributions of mergers with different properties (e.g., mass ratios, gas fractions, redshifts) to the bulge population. We compare with observational constraints, and find good agreement. We also provide useful fitting functions and make public a code to reproduce the predicted merger rates and contributions to bulge mass growth. We identify several robust conclusions. (1) Major mergers dominate the formation and assembly of ~L * bulges and the total spheroid mass density, but minor mergers contribute a non-negligible ~30%. (2) This is mass dependent: bulge formation and assembly is dominated by more minor mergers in lower-mass systems. In higher-mass systems, most bulges originally form in major mergers near ~L *, but assemble in increasingly minor mergers. (3) The minor/major contribution is also morphology dependent: higher B/T systems preferentially form in more major mergers, with B/T roughly tracing the mass ratio of the largest recent merger; lower B/T systems preferentially form in situ from minor mergers. (4) Low-mass galaxies, being gas-rich, require more mergers to reach the same B/T as high-mass systems. Gas-richness dramatically suppresses the absolute efficiency of bulge formation, but does not strongly influence the relative contribution of major versus minor mergers. (5) Absolute merger rates at fixed mass ratio increase with galaxy mass. (6) Predicted merger rates agree well with those observed in pair and morphology-selected samples, but there is evidence that some morphology-selected samples include contamination from minor mergers. (7) Predicted rates also agree with the integrated growth in bulge mass density with cosmic time, but with a factor ~2 uncertainty in both—up to half the bulge mass density could come from non-merger processes. We systematically vary the model assumptions, totaling ~103 model permutations, and quantify the resulting uncertainties. Our conclusions regarding the importance of different mergers for bulge formation are very robust to these changes. The absolute predicted merger rates are systematically uncertain at the factor ~2 level; uncertainties grow at the lowest masses and high redshifts.
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