Can Muslims make Du`a' (supplication) to Allah using His Names for the cure of their illnesses?
It is permissible to do so due to the general saying of Allah (Exalted be He):
And (all) the Most Beautiful Names belong to Allâh, so call on Him by them
This is also authentically reported from the Prophet (peace be upon him). He (peace be upon him) pronounced Ruqyah (reciting Qur'an and saying supplications over the sick seeking healing) by saying:
O Lord of the people! Remove the difficulty and bring about healing as You are the Healer. There is no healing but Your Healing.
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.