Is it permissible, in Islamic law, to adopt the method of treating toothache by engraving seven Arabic letters on the ground, placing a nail on each letter while reciting verses from the Qur’an, and then hammering the nail into the ground when the pain subsides?

There is no objection to seeking treatment for illnesses, including toothache. Combining medical treatment with legitimate Ruqyah (definition) is better. The mentioned remedies for pain relief, with the exception of the Quran, are superstitions with no basis. The means used for treatment and Ruqyah must be either legally prescribed (those proven beneficial by Islamic law) or empirically proven (those proven beneficial by experience). Relying on something that is neither a legally prescribed nor an empirically proven means is shirk (definition) (polytheism), such as using disjointed letters, which is a method employed by sorcerers and soothsayers.