We present an ab initio theory and computational method for Penning ionization widths. Our method is based on the Fano theory of resonances, algebraic diagrammatic construction (ADC) scheme for many-electron systems, and Stieltjes imaging procedure. It includes an extension of the Fano-ADC scheme [V. Averbukh and L. S. Cederbaum, J. Chem. Phys. 123, 204107 (2005)] to triplet excited states. Penning ionization widths of various He*-H2states are calculated as a function of the distance R between He* and H2. We analyze the asymptotic (large-R) dependences of the Penning widths in the region where the well-established electron transfer mechanism of the decay is suppressed by the multipole- and/or spin-forbidden energy transfer. The R-12and R-8power laws are derived for the asymptotes of the Penning widths of the singlet and triplet excited states of He*(1s2s1,3S), respectively. We show that the electron transfer mechanism dominates Penning ionization of He*(1s2s3S)-H2up until the He*-H2separation is large enough for the radiative decay of He* to become the dominant channel. The same mechanism also dominates the ionization of He*(1s2s1S)-H2when R < 5 Å. We estimate that the regime of energy transfer in the He*-H2Penning ionization cannot be reached by approaching zero collisional temperature. However, the multipole-forbidden energy transfer mechanism can become important for Penning ionization in doped helium droplets.