There are a number of verses in the Qur'aan that are used as evidence that the Qur'aan can be used as a healing for physical illnesses. Among these verses are:
"And We send down from the Qur'aan that which is a healing (shifaa') and a mercy (rahma) to those who believe. And it increases the Dhaalimeen (polytheists and wrong-doers) nothing but loss." (Sooratul-Israa', 17:82)
"Say: 'It is for those who believe, a guide (huda) and a healing (shifaa').'" (Soorah Fussilat, 41:44)
There is some discussion about whether these verses refer to physical diseases or only spiritual diseases. 
In any case, the following hadeeth is clear that the Qur'aan is a cure for physical as well as spiritual ailments: Sayyiduna Abu Sa'eed al-Khudree رضي الله عنه narrated that some of the Companions رضي الله عنهم of the Prophet صلى الله عليه وسلم were among an Arab tribe that refused to be hospitable toward them. While they were among that tribe, the chief of the tribe was bitten by a snake (or stung by a scorpion) and they asked the Companions رضي الله عنهم, "Have you got any medicine with you or anyone who can make an incantation (ruqya)?" They answered, "You refused to be hospitable toward us so we will not treat him unless you pas us for it." So they agreed to pay them a flock of sheep. One of the Companions رضي الله عنهم started reciting Sooratul-Faatihah and gathering his saliva and spitting it onto the place of the bite. The chief was cured and his people presented the sheep to the Companions رضي الله عنهم but they said (to themselves), "We will not take it until we ask the Prophet صلى الله عليه وسلم (if it is lawful for us)." When they asked him he smiled and said, "How did you know that Sooratul-Faatihah is an an incantation (ruqya)? Take it (the flock of sheep) and assign me a share." 
But obviously, if this is the only way that a person uses the Qur'aan or turns to it, then he will be missing the greater purpose of the Qur'aan. He will be using it for something for which it may be used, healing physical illnesses, but he will not be using it for something which is much more vital to his welfare: the healing of spiritual illnesses. Allaah سبحانه و تعالى says,
"O mankind! There has come to you a good advice from your Lord (i.e. the Qur'aan, ordering all that is good and forbidding all that is evil), and a healing (shifaa') for that (disease of ignorance, doubt, hypocrisy and differences, etc.) in your breasts, - a guidance (huda) and a mercy (rahma) (explaining lawful and unlawful things, etc.) for the believers." (Soorah Yoonus: 10:57)
Commenting on this verse, Shaykh 'Abdul-Hameed Ibn Baadees رحمه الله wrote,
"This verse in Soorah Yoonus specifies its healing as being for what is in the hearts, which is where the beliefs reside. This is because that is the main goal of the Qur'aan and it is the foundation for other aspects. This is because if the hearts are healed of evil beliefs and the doubts are extracted from them and the truth is believed and held with certainty, the souls are then purified and the behaviour of the person, both as an individual and his society, will be set aright and they will be ascending the steps fo completeness.
This fact is not negated by the Qur'aan also being a cure for evil character, as its being mentioned in a general sense in the verses [quoted above] from Sooratul-Israa' and Sooratus-Sajdah would indicate. This is because character originates with beliefs and is a necessary consequence thereof. Furthermore, the soul of humans is not perfected until both of them are healed. This also does not negate the fact that the Qur'aan, on some occasions, may be a healing for physical ailments, as the [unrestricted mention of] healing would also imply... However, that is not the main purpose of what is meant by the healing of the Qur'aan." 
Shaykh 'Abdul-Hameed Ibn Baadees رحمه الله goes on to explain that illnesses are either spiritual or physical. The spiritual diseases are those of the mind and soul. The diseases related to the mind include apathy, misconceptions, blind following of the cultures and forefathers' ways, false beliefs and doubts about the truth. The diseases related to the soul include bad character, lusts and evil desires.
The Qur'aan presents a cure for both of these types of spiritual diseases. With respect to the first type, the Qur'aan invites and requests mankind to think, ponder, understand and investigate themselves and the creation around them. The Qur'aan shows mankind how to perceive the reality of this creaton and how to think. It blames those who blindly follow the teachings of their forefathers even when their forefathers were in errant. Then the Qur'aan presents convincing and unquestionable arguments concerning the reality of this world. The end result for those who wish to be cured of such illnesses will be a strong belief in the truth with no doubts or confusion.
As for the second type of spiritual diseases, the diseases of the soul, the Qur'aan reminds and presents to the person the evil results of such diseases. At the same time, it guides one to the proper behaviour and character, showing that they are of great benefit and importance.
Thi is how the spiritual diseases are cured. They both go back, though, to curing the beliefs, which is the foundation for the other realms. In fact, the only real cure for such diseases can be found in the Qur'aan. If one looks for their cures in any other sources, then the disease will only increase. 
Such in fact is the case nowadays as people turn to man-made theories and ideologies to find cures for problems and diseases for which only the Creator سبحانه و تعالى can offer a true and complete cure. 
 For such a discussion see Muhammad Ibraaheem Saleem, At-Tadaawi bil-Qur'aan wal-Istishfaa bir-Ruqa wat-Ta'weedh (Cairo: Maktabah al-Qur'aan, n.d.), pp. 72-82. On pp. 137-144 he discusses the experiments with the Qur'aan as a source of healing that were performed in Panama City, Florida.
 Recorded by Imaam al-Bukhaaree رحمه الله in his Saheeh.
 Tafseer Ibn Baadees fee Majaalis at-Tadhkeer min Kalaam al-Hakeem al-Khabeer (Beirut: Daarul-Kutub al-'Ilmiyyah, 1995), p. 143.
 Tafseer Ibn Baadees fee Majaalis at-Tadhkeer min Kalaam al-Hakeem al-Khabeer (Beirut: Daarul-Kutub al-'Ilmiyyah, 1995), pp. 143-144. Also see 'Abdullaah an-Noori, al-Urwatul-Wuthqa (Beirut: Daarul-'Arabiyyah, n.d.), pp. 43-46.
 How to Approach and Understand the Qur'aan, pp. 68-71.