We present post-Newtonian $N$-body simulations on mergers of accreting stellar-mass black holes (BHs), where such general relativistic effects as the pericenter shift and gravitational wave (GW) emission are taken into consideration. The attention is concentrated on the effects of the dynamical friction and the Hoyle-Lyttleton mass accretion by ambient gas. We consider a system composed of ten BHs with initial mass of $30~M_odot$. As a result, we show that mergers of accreting stellar-mass BHs are classified into four types: a gas drag-driven, an interplay-driven, a three body-driven, or an accretion-driven merger. We find that BH mergers proceed before significant mass accretion, even if the accretion rate is $sim10$ Eddington accretion rate, and then all BHs can merge into one heavy BH. Using the simulation results for a wide range of parameters, we derive a critical accretion rate ($dot{m}_{rm c}$), below which the BH growth is promoted faster by mergers. Also, it is found that the effect of the recoil by the GW emission can reduce $dot{m}_{rm c}$ especially in gas number density higher than $10^8~{rm cm}^{-3}$, and enhance the escape probability of merged BHs. Very recently, a gravitational wave event, GW150914, as a result of the merger of a $sim 30~M_odot$ BH binary has been detected (Abbott et al. 2016). Based on the present simulations, the BH merger in GW150914 is likely to be driven by three-body encounters accompanied by a few $M_odot$ of gas accretion, in high-density environments like dense interstellar clouds or galactic nuclei.
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