In this paper, we carry out the complete $mathcal O(alphaalpha_s^{2})$-order study on the inclusive productions of $Upsilon(nS)$ and $chi_b(nP)$ ($n=1,2,3$) via the Standard Model Higgs boson decay, within the framework of nonrelativistic QCD. The feeddown effects via the higher excited states are found to be substantial. The color-octet $^3S_1^{[8]}$ state related processes consisting of $H^0 to bbar{b}[^3S_1^{[8]}]+g$ and $H^0 to bbar{b}[^3S_1^{[8]}]+Q+bar{Q}$ ($Q=c,b$) play a vital role in the predictions on the decay widths. Moreover, our newly calculated next-to-leading order QCD corrections to $H^0 to bbar{b}[^3S_1^{[8]}]+g$ can enhance its leading-order result by 3-4 times, subsequently magnifying the total $^3S_1^{[8]}$ contributions by about $40%$. Such a remarkable enhancement will to a large extent influence the phenomenological conclusions. For the color-singlet $^3P_J^{[1]}$ state, in addition to $H^{0} to bbar{b}[^3P_J^{[1]}]+b+bar{b}$, the newly introduced light hadrons associated process, $H^{0} to bbar{b}[^3P_J^{[1]}]+g+g$, can also provide non-negligible contributions, especially for $^3P_2^{[1]}$. Summing up all the contributions, we have $mathcal B_{H^0 to chi_b(nP)+X} sim 10^{-6}-10^{-5}$ and $mathcal B_{H^0 to Upsilon(nS)+X} sim 10^{-5}-10^{-4}$, which meets marginally nowadays LHC experimental data and can help in understanding the heavy quarkonium production mechanism as well as the Yukawa couplings.
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