Short answer
Islam teaches that all healing comes from God (﷾), and that He heals through means. So a sick Muslim does both: he seeks treatment, and he prays. The Prophet (ﷺ) said Allah (﷾) has not sent down any illness without sending down its cure. When people asked if they should seek treatment, he said yes. He also prayed over the sick. He asked the Lord of the people to remove the harm, for He is the Healer. His Companions recited the Qur'an's opening chapter over a sick man, and he approved. He taught words to say over pain, three times and seven times. When he was ill, he recited the Qur'an's last two chapters over himself. The Qur'an calls itself a healing, and says of honey that in it "is healing for men". Illness borne with patience also wipes away sins. So a Muslim needs no showman who claims special powers. He goes to a doctor, prays, recites the Qur'an, and trusts in God (﷾).
Full explanation
Healing belongs to God (﷾)
The Qur'an gives Abraham's words: "And when I am ill, it is He Who cures me". Doctors, medicine and prayer are means. God (﷾) is the One who makes them work. The Prophet (ﷺ) prayed over the sick in the same spirit. He asked the Lord of the people to remove the harm and heal, for He is the Healer. There is no healing but His (Bukhari 5675).
“"And when I am ill, it is He Who cures me;”
Aisha reports that when the Prophet (ﷺ) visited a sick person, he prayed: remove the harm, Lord of the people, and heal, for You are the Healer; there is no healing but Yours, a healing that leaves no illness behind.
Seek treatment
The Prophet (ﷺ) said Allah (﷾) has not sent down any illness without sending down its cure (Bukhari 5678). Some Bedouins asked him whether they should seek treatment. He told them to seek it, for Allah (﷾) made a cure for every illness except old age (Abu Dawud 3855). So going to a doctor is not weak faith. It is following the Prophet (ﷺ).
Abu Hurayra reports that the Prophet (ﷺ) said God has not sent down any disease without also sending down its cure.
Usama ibn Sharik reports that desert Arabs asked the Prophet (ﷺ) whether they should seek medical treatment. He told them to seek treatment, for God has not made any disease without making a remedy for it, except one: old age.
Pray for the sick, and for yourself
The Qur'an tells of the prophet Job, who suffered long. He called on his Lord: "Truly distress has seized me, but Thou art the Most Merciful of those that are merciful." God (﷾) answered him and removed his distress. A Muslim prays for his own healing and for others. He visits the sick and prays over them, as the Prophet (ﷺ) did.
“And (remember) Job, when He cried to his Lord, "Truly distress has seized me, but Thou art the Most Merciful of those that are merciful." So We listened to him: We removed the distress that was on him, and We restored his people to him, and doubled their number,- as a Grace from Ourselves, and a thing for commemoration, for all who serve Us.”
Aisha reports that when the Prophet (ﷺ) visited a sick person, he prayed: remove the harm, Lord of the people, and heal, for You are the Healer; there is no healing but Yours, a healing that leaves no illness behind.
The Qur'an as healing
The Qur'an says: "We send down (stage by stage) in the Qur'an that which is a healing and a mercy to those who believe". It calls itself "a healing for the (diseases) in your hearts". Its first healing is of the heart: worry, fear and despair. Reciting it over the sick is called ruqya. Some Companions recited the Fatiha, the Qur'an's opening chapter, over a man who had been stung, and he recovered. The Prophet (ﷺ) approved (Bukhari 5736).
“We send down (stage by stage) in the Qur'an that which is a healing and a mercy to those who believe: to the unjust it causes nothing but loss after loss.”
“O mankind! there hath come to you a direction from your Lord and a healing for the (diseases) in your hearts,- and for those who believe, a guidance and a Mercy.”
Abu Sa'id reports that some Companions passed a tribe whose chief had been stung. One of them recited the Fatiha (the Qur'an's opening chapter) over him, and he recovered. When they told the Prophet (ﷺ), he asked how they knew it was a ruqya (a healing recitation), and approved.
Words of healing the Prophet (ﷺ) taught
A man complained to the Prophet (ﷺ) of a pain in his body. He told him to put his hand where it hurt and say "Bismillah" three times. Then seven times: "I seek refuge in Allah and His power from the evil of what I feel and fear" (Muslim 2202). For visiting the sick, he taught a prayer to say seven times beside them. It is: "I ask Allah the Mighty, Lord of the Mighty Throne, to heal you". He said Allah (﷾) will heal the sick person, unless his time has come (Abu Dawud 3106). When the Prophet (ﷺ) himself was ill, the angel Gabriel recited over him. He said: "In the name of Allah I recite over you", from everything that harms you; may Allah (﷾) heal you (Muslim 2186).
Uthman ibn Abi al-As complained to the Prophet (ﷺ) of a pain he had felt in his body since becoming Muslim. The Prophet (ﷺ) told him to put his hand on the place of the pain and say Bismillah (in the name of Allah) three times. Then seven times: A'udhu billahi wa qudratihi min sharri ma ajidu wa uhadhir, "I seek refuge in Allah and His power from the evil of what I feel and fear".
Ibn Abbas reports that the Prophet (ﷺ) said: whoever visits a sick person whose time has not come, and says seven times beside him, As'alullah al-'Azim, Rabb al-'Arsh al-'Azim, an yashfiyak, "I ask Allah the Mighty, Lord of the Mighty Throne, to heal you", Allah will heal him of that illness.
Abu Sa'id reports that Gabriel came to the Prophet (ﷺ) and asked if he was ill. He said yes. Gabriel said: Bismillahi arqik, "In the name of Allah I recite over you", from everything that harms you, from the evil of every soul or envious eye; may Allah heal you; in the name of Allah I recite over you.
The last chapters of the Qur'an
Whenever the Prophet (ﷺ) was ill, he recited the last two chapters of the Qur'an over himself and blew lightly. They are al-Falaq and al-Nas, called the two chapters of refuge. In his last illness, when he was too weak, his wife Aisha recited them over him (Bukhari 5016). Every night before sleep, he recited al-Ikhlas, al-Falaq and al-Nas into his cupped hands. Then he wiped his hands over his body, three times (Bukhari 5017). Al-Falaq asks refuge "from the mischief of created things".
Aisha reports that whenever the Prophet (ﷺ) was ill, he recited al-Mu'awwidhat (Surat al-Falaq and Surat al-Nas) over himself and blew lightly. When his pain grew severe, she recited them over him and wiped his own hand over him, hoping for its blessing.
Aisha reports that every night, when the Prophet (ﷺ) went to bed, he cupped his hands together, recited Surat al-Ikhlas, Surat al-Falaq and Surat al-Nas, and blew into them. Then he wiped his hands over as much of his body as he could, beginning with his head, his face and the front of his body. He did this three times.
“Say: I seek refuge with the Lord of the Dawn From the mischief of created things; From the mischief of Darkness as it overspreads; From the mischief of those who practise secret arts; And from the mischief of the envious one as he practises envy.”
See a doctor
Allah (﷾) sent down a cure for every illness.
Pray to God (﷾)
Ask the Lord of the people to remove the harm, for He is the Healer.
Hand on the pain
Say Bismillah three times. Then seven times: I seek refuge in Allah (﷾) and His power from the evil of what I feel and fear.
Recite the Qur'an
The Fatiha; and al-Ikhlas, al-Falaq and al-Nas into the hands, then wipe over the body, three times.
Visit the sick
Seven times: I ask Allah (﷾) the Mighty, Lord of the Mighty Throne, to heal you.
Be patient
Illness borne with patience wipes away sins.
Only words that call on God (﷾)
Before Islam, the Arabs had their own healing chants. They asked the Prophet (ﷺ) about them. He asked to hear them. He said they are allowed as long as they hold no shirk: the worship of anything beside God (Muslim 2200). So a Muslim may recite the Qur'an and pray to God (﷾) for healing. He may not call on spirits, saints or charms. Nor does he need anyone with "special powers". Any Muslim may recite these words over himself or someone he loves.
'Awf ibn Malik al-Ashja'i reports that before Islam they used words of healing (ruqya), and they asked the Prophet (ﷺ) about them. He said: show me your words of healing; there is no harm in them as long as they hold no shirk (worship of anything beside God).
Honey and black seed
The Qur'an says of the bee: "there issues from within their bodies a drink of varying colours, wherein is healing for men". A man told the Prophet (ﷺ) his brother had a stomach complaint. He told him to give his brother honey, and the brother recovered (Bukhari 5684). He also said there is healing in black seed for every disease except death (Bukhari 5688). Honey and black seed are among the means God (﷾) has given. They are not a reason to refuse a doctor's treatment.
“Then to eat of all the produce (of the earth), and find with skill the spacious paths of its Lord: there issues from within their bodies a drink of varying colours, wherein is healing for men: verily in this is a Sign for those who give thought.”
Abu Sa'id al-Khudri reports that a man told the Prophet (ﷺ) his brother had a stomach complaint, and the Prophet told him to give him honey. The man came back more than once saying it had not worked, and each time the Prophet said to give him honey, adding that Allah had spoken the truth and it was the brother's stomach that was wrong. He drank honey again and was cured.
Abu Hurayra reports that he heard the Prophet (ﷺ) say there is healing in black seed (nigella) for every disease except death.
Usama ibn Sharik reports that desert Arabs asked the Prophet (ﷺ) whether they should seek medical treatment. He told them to seek treatment, for God has not made any disease without making a remedy for it, except one: old age.
Patience in illness
The Prophet (ﷺ) said no illness, worry or harm reaches a Muslim without wiping away some of his sins. Even a thorn's prick does (Bukhari 5641). The Qur'an gives "glad tidings to those who patiently persevere" when trials come. So illness is never wasted. The sick person seeks a cure, and while he waits, his patience is rewarded.
Abu Sa'id al-Khudri and Abu Hurayra report that the Prophet (ﷺ) said no tiredness, illness, sorrow, sadness, hurt or distress reaches a Muslim, not even the prick of a thorn, without Allah wiping away some of his sins because of it.
“Be sure we shall test you with something of fear and hunger, some loss in goods or lives or the fruits (of your toil), but give glad tidings to those who patiently persevere, Who say, when afflicted with calamity: "To Allah We belong, and to Him is our return":- They are those on whom (Descend) blessings from Allah, and Mercy, and they are the ones that receive guidance.”
Perspectives
Only medicine heals
Many secular people say only medicine heals, and that prayer or reciting scripture over the sick does nothing.
Only medicine heals; prayer does nothing for the body.
Rebuttal
Islam does not ask anyone to choose. The Prophet (ﷺ) told people to seek treatment, and also prayed for the sick. Medicine is a means God (﷾) created. Prayer asks the One who made the means to make them work, and it brings calm and hope.
The Islamic perspective
All healing comes from God (﷾). A Muslim seeks treatment, prays, recites the Qur'an and bears illness with patience, trusting God (﷾).
Allah (﷾) has not sent down an illness without sending down its cure.
The Prophet (ﷺ) told people to seek treatment.
The Prophet (ﷺ) prayed over the sick for healing.
The Qur'an calls itself a healing and a mercy to believers.
Illness borne with patience wipes away sins.
The Prophet (ﷺ) taught: put your hand on the pain, say Bismillah three times, and seek refuge in Allah (﷾) seven times.
When ill, the Prophet (ﷺ) recited al-Falaq and al-Nas over himself.
Words of healing are allowed as long as they hold no shirk.