What is the ruling of Islamic law regarding the custom of bathing a child with water mixed with dirt that has been placed under the stars for healing from illness?

What you have asked about is from the realm of illusions and superstitions that have no basis in Islamic law or empirical medicine. A Muslim must avoid it. Islamic law has urged taking direct action, including seeking medical treatment. The Prophet, peace and blessings be upon him, said: "Seek medical treatment, for Allah Almighty has not created a disease without creating a cure for it, except for one disease: old age." Treatment should be with medicines whose benefit has been proven by Islamic law or medically. The means that Allah has made causes are of two types: legitimate, like supplication (du'a), and tangible, like known physical medicines, whether through revelation or experience. As for mere illusions and fantasies, it is not permissible to rely on them or assert that they are a cure. Furthermore, this behavior resembles the actions of idolaters who worship stars and planets, and it causes the heart to become attached to other than Allah, which weakens the faith of its doer.