Recent LHC data show hints of a new resonance in the diphoton distribution at an invariant mass of 750 GeV. Interestingly, this new particle might be both CP odd and play the role of a portal into the dark matter sector. Under these assumptions and motivated by the fact that the requirement of SU(2) L invariance automatically implies the coupling of this alleged new resonance to ZZ and Z γ, we investigate the current and future constraints coming from the indirect searches performed through the neutrino telescope IceCube, supplementing them with direct detection experiments and γ-ray observations. We show that IceCube constraints can be stronger than the ones from direct detection experiments if the dark matter mass is larger than a few hundred GeV. Furthermore, in the scenario in which the dark matter is a scalar particle, the IceCube data limit the cross section between the DM and the proton to values close to the predicted ones for natural values of the parameters.