Calculations at the (14,10)CASSCF/6-31G** and the MR-(S)DCI/cc-pVTZ levels are employed to answer the title question by studying three possible modes of reaction between dioxygen and dihydrogen molecules at the ground triplet state and excited singlet state of O-2. These reaction modes, which are analogous to well-established mechanisms for oxidants such as transition metal oxene cations and mono-oxygenase enzymes, are the following: (i) the concerted addition, (ii) the oxene-insertion, and (iii) the hydrogen abstraction followed by hydrogen rebound. The "rebound" mechanism is found to be the most preferable of the three mechanisms. However, the barrier of the H-abstraction step is substantial both for the triplet and the singlet states of O-2, and the process is highly endothermic (>30 kcal/mol) and is unlikely to proceed at ambient conditions. The calculations revealed also that the lowest singlet state of O-2 has very high barriers for reaction and therefore cannot mediate a facile oxidation of H-2 in contrast to transition metal oxenide cation catalysts and monooxygenase enzymes. This fundamental difference is explained.