Is saying that a patient is beyond cure considered despairing of God's mercy, and is it permissible for us to say that?

This statement is incorrect, and it is not permissible for a Muslim to utter it. It contains despair from the mercy of Allah. Rather, one should continue to supplicate for healing, recite ruqyah (Islamic incantations), and seek treatment, for Allah is the Healer, capable of all things.