When Hajime Nagumo ate a wolf-like monster, he started to feel sick and/or die, but when he drank the sacred water he started to heal. Is this why Hajime Nagumo can eat monsters or can anyone eat monsters
Nagumo was indeed able to survive by drinking the secretions of the stone. Which in later parts of the manga we find out is actually.
Ambrosia, a liquid secreted from the Divinity stone that heals all wounds. (except dismembered limbs)
The consumption of the wolf-like monster started the destruction of his body. Shattering his bones, destroying his organs, which would kill every normal human if not constantly healed by an external source.
Besides Nagumo, there is only one person likely able to survive the consumption of the wolf-like monster. Which would be:
Yue, A vampire with the ability to regenerate all damage, that does not instantly vaporize her.
All though the limits of her regeneration are not yet known, and thus might also be unable to consume it.
Initially, yes. Though consuming monsters should have killed him, the powerful potion negated their toxicity and healed his injuries.
Later on, however, a constant diet of monster flesh and Ambrosia changes Hajime's body to the point where he can eat most monsters without even feeling it. Only the strongest or most toxic monsters have any negative effect on him anymore. Even then it is more uncomfortable than dangerous.