We compute the $S$-factor of the proton-proton ($pp$) fusion reaction using chiral effective field theory ($chi$EFT) up to next-to-next-to-leading order (NNLO) and perform a rigorous uncertainty analysis of the results. We quantify the uncertainties due to (i) the computational method used to compute the $pp$ cross section in momentum space, (ii) the statistical uncertainties in the low-energy coupling constants of $chi$EFT, (iii) the systematic uncertainty due to the $chi$EFT cutoff, and (iv) systematic variations in the database used to calibrate the nucleon-nucleon interaction. We also examine the robustness of the polynomial extrapolation procedure, which is commonly used to extract the threshold $S$-factor and its energy-derivatives. By performing a statistical analysis of the polynomial fit of the energy-dependent $S$-factor at several different energy intervals, we eliminate a systematic uncertainty that can arise from the choice of the fit interval in our calculations. In addition, we explore the statistical correlations between the $S$-factor and few-nucleon observables such as the binding energies and point-proton radii of $^{2,3}$H and $^3$He as well as the $D$-state probability and quadrupole moment of $^2$H, and the $beta$-decay of $^{3}$H. We find that, with the state-of-the-art optimization of the nuclear Hamiltonian, the statistical uncertainty in the threshold $S$-factor cannot be reduced beyond 0.7%.
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