Observation of gravitationally induced entanglement between two massive particles is the evidence of quantum nature of gravity. However, quantum effects in the gravitational field are exceptionally small so that the existing quantum-gravity proposals are extremely hard to test in practice. For addressing this key challenge, here we propose a criterion based on the logical contradictions of weak entanglement, which may boost the sensitivity of the signal due to the gravitationally induced entanglement. Specifically, we make use of the weak-value scenario and Einstein-Podolsky-Rosen (EPR) steering. We prove that it is impossible for a classical mediator to act on two local quantum objects to simulate amplified-weak-value phenomenon in two-setting EPR steering. Compared to the previous protocols, our approach can amplify the signal of quantum gravity by any desired factor that depends on the magnitude of the weak value. Our results not only open up the possibility of detecting quantum gravity in the near future, but also pave the way for weak entanglement criterion of a more general nature.