Short answer
No. The Prophet Muhammad (ﷺ) could not read or write. The Qur'an said so to the people who had known him all his life, and none of them answered that he could read. The Qur'an speaks as God, in the first person. Hundreds of times it tells him what to say, and it even corrects him in public. It does not sound like his own recorded words. When his wife Aisha was slandered, he waited a month for revelation instead of making up a verse to clear her. Abu Sufyan, a leader of Mecca and then still his enemy, said he had never been accused of lying before his mission. When his opponents asked him to soften the message or change the Qur'an, he refused, and he died poor. Islam's answer is that the Qur'an is the word of God (﷾), brought to him by the angel Gabriel.
Full explanation
A man who could not read or write
The Qur'an reminds the Prophet (ﷺ) that he never read a book or wrote one with his hand. It says that, had he done so, those who deal in falsehood would have had grounds to doubt. It calls him the unlettered Prophet (ﷺ), meaning one who cannot read or write. These words were recited to his own townspeople, who had known him for forty years. No one answered that he could read. His enemies said he took his stories from others. Even they said that these were written down for him and read out to him. The Qur'an answered that it was sent down by Him who knows the mystery in the heavens and the earth. When the angel first told him to read, he replied that he could not. Waraqa, a Christian cousin of his wife Khadija, is the only Christian named in the account of the first revelation. Waraqa died a few days after the first revelation.
“And thou wast not (able) to recite a Book before this (Book came), nor art thou (able) to transcribe it with thy right hand: In that case, indeed, would the talkers of vanities have doubted.”
“Those who follow the messenger, the unlettered Prophet, whom they find mentioned in their own (scriptures),- in the law and the Gospel;-”
“And they say: "Tales of the ancients, which he has caused to be written: and they are dictated before him morning and evening."”
“Say: "The (Qur'an) was sent down by Him who knows the mystery (that is) in the heavens and the earth: verily He is Oft-Forgiving, Most Merciful."”
Aisha describes the first revelation. In the cave of Hira the angel told the Prophet (ﷺ) to read, and three times he answered that he could not read, before the first verses of Surat al-Alaq were given to him. Khadija took him to her cousin Waraqa ibn Nawfal, an old, blind man who had become a Christian and wrote out parts of the Gospel in Hebrew letters. Waraqa said this was the angel God had sent to Moses, and a few days later he died.
God (﷾) speaks in the first person
In many verses God (﷾) speaks as "I" and "Me". Some even speak to the Prophet (ﷺ) as "thee" about God's (﷾) servants, with no "God said" before them. No Muslim ever claimed he was God, so these are not his words about himself. Hundreds of verses contain the command "Say", most of them addressed to him. He is the one being told what to say, not the one composing it.
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
“I have only created Jinns and men, that they may serve Me.”
“Verily, this brotherhood of yours is a single brotherhood, and I am your Lord and Cherisher: therefore serve Me (and no other).”
“But when Our Clear Signs are rehearsed unto them, those who rest not their hope on their meeting with Us, Say: "Bring us a reading other than this, or change this," Say: "It is not for me, of my own accord, to change it: I follow naught but what is revealed unto me: if I were to disobey my Lord, I should myself fear the penalty of a Great Day (to come)."”
Three kinds of speech that were never mixed
The recorded sayings of the Prophet (ﷺ) are the speech of a man: direct, practical, answering the question in front of him. The Qur'an is different. It has its own rhythm, and often one rhyme that runs through a whole chapter, unlike his everyday speech. It also speaks to him from outside. His own words were kept in separate collections, the hadith, which are reports of what he said and did. They never entered the Qur'an. A third kind, the hadith qudsi, gives God's (﷾) meaning in the Prophet's (ﷺ) own words. It too was kept out of the Qur'an from the start. The Bible, by contrast, carries its writers' personal greetings to their friends inside its text.
The Prophet (ﷺ) reports that Allah says: I am as My servant thinks of Me, and I am with him when he remembers Me; if he comes a hand-span nearer to Me, I come an arm's length nearer to him, and if he comes to Me walking, I go to him running. Also in Sahih Muslim 2675.
“Salute Prisca and Aquila, and the household of Onesiphorus.”
“All that are with me salute thee. Greet them that love us in the faith. Grace be with you all. Amen.”
The Qur'an came through the Prophet Muhammad (ﷺ). Did he write it himself?
Could he read or write?
No. The Qur'an says he never read or wrote a book before it came to him.
Do the words sound like his?
No. The Qur'an speaks as God, saying “I” and “Me”, and tells him what to “Say”.
Would he correct himself in public?
The Qur'an sometimes corrects him, and he recited those verses aloud all his life.
Could he bring a verse when he needed one?
No. When lies were spread about his wife Aisha, he waited a month for the verses that cleared her.
Was he a liar?
No. His enemy Abu Sufyan admitted that before his mission no one had known him to lie. A man who did not lie about people would not lie about God (﷾).
Did he do it for money?
No. He died so poor that his armour was held as security for barley.
So the Qur'an is not his work. It is the word of God (﷾), passed on as he received it.
The Book corrects him
The Qur'an records that the Prophet (ﷺ) frowned and turned away from a blind man who came to learn. It corrects him for it. Another verse asks him why he forbade himself what Allah (﷾) had made lawful to him. He had to recite these corrections aloud for the rest of his life. A man writing a book to raise his own standing does not put his own rebuke in it. Nor did he take praise he was offered. His son Ibrahim died on the day of an eclipse, and people said the sun had darkened for him. He told them eclipses happen for no one's death or birth.
“(The Prophet) frowned and turned away, Because there came to him the blind man (interrupting). But what could tell thee but that perchance he might grow (in spiritual understanding)?- Or that he might receive admonition, and the teaching might profit him? As to one who regards Himself as self-sufficient, To him dost thou attend; Though it is no blame to thee if he grow not (in spiritual understanding). But as to him who came to thee striving earnestly, And with fear (in his heart), Of him wast thou unmindful.”
“O Prophet! Why holdest thou to be forbidden that which Allah has made lawful to thee? Thou seekest to please thy consorts. But Allah is Oft-Forgiving, Most Merciful.”
Al-Mughira ibn Shu'ba reports that the sun eclipsed on the day the Prophet's (ﷺ) son Ibrahim died, and people said it had eclipsed because of Ibrahim's death. The Prophet said that the sun and the moon do not eclipse for the death or the birth of anyone, and told them to pray and call on Allah when they see an eclipse.
A month of silence
Aisha, the wife of the Prophet (ﷺ), was slandered, meaning falsely accused. The rumour ran through the city of Medina. In her own words, he went a whole month without any revelation about her case. Only then did the verses declaring her innocence come. A man composing scripture would have cleared his wife at once.
Aisha's own account of the slander spread against her. She says the Prophet (ﷺ) went a whole month without receiving any revelation about her case, while the rumour spread and he consulted his Companions, before the verses declaring her innocence came down. When Safwan ibn al-Mu'attal found her left behind and knew her, for he had seen her before the veil was ordained, she says: I covered my face with my garment.
“Those who brought forward the lie are a body among yourselves: think it not to be an evil to you; On the contrary it is good for you: to every man among them (will come the punishment) of the sin that he earned, and to him who took on himself the lead among them, will be a penalty grievous.”
What his enemies said
Heraclius, the Roman emperor, asked Abu Sufyan whether Muhammad (ﷺ) had ever been accused of lying before he claimed to be a prophet. Abu Sufyan was still an enemy of the Prophet (ﷺ) then. He said no. His opponents wanted him to bend a little so that they could bend too. When they asked for a different Qur'an, he was told to answer that he could not change it of his own accord. He died with his armour pawned for barley.
Abu Sufyan, at the time still an enemy of the Prophet (ﷺ), was questioned by the Byzantine emperor Heraclius. Asked whether they had ever accused Muhammad of lying before he claimed prophethood, he answered no. Heraclius concluded that a man who did not lie about people would not lie about God.
“Their desire is that thou shouldst be pliant: so would they be pliant.”
“But when Our Clear Signs are rehearsed unto them, those who rest not their hope on their meeting with Us, Say: "Bring us a reading other than this, or change this," Say: "It is not for me, of my own accord, to change it: I follow naught but what is revealed unto me: if I were to disobey my Lord, I should myself fear the penalty of a Great Day (to come)."”
Aisha reports that when the Prophet (ﷺ) died, his armour was held by a Jewish man as security for thirty measures of barley.
Stories his people did not know
After the story of Noah the Qur'an says that neither the Prophet (ﷺ) nor his people knew these stories before. Some of the details it gives are in no earlier scripture. Noah's son, who refused the ark and drowned, is found nowhere in the Bible.
“Such are some of the stories of the unseen, which We have revealed unto thee: before this, neither thou nor thy people knew them. So persevere patiently: for the End is for those who are righteous.”
“So the Ark floated with them on the waves (towering) like mountains, and Noah called out to his son, who had separated himself (from the rest): "O my son! embark with us, and be not with the unbelievers!" The son replied: "I will betake myself to some mountain: it will save me from the water." Noah said: "This day nothing can save, from the command of Allah, any but those on whom He hath mercy! "And the waves came between them, and the son was among those overwhelmed in the Flood.”
Perspectives
What the historical record shows
The earliest sources describe the Prophet (ﷺ) as a man known for honesty. He did not grow rich from his message, and he died with his armour pawned for barley. He could not produce revelation at will, even when he most needed it.
Aisha reports that the Prophet (ﷺ) received no revelation about the slander against her for a month.
Abu Sufyan was still his enemy when he spoke to Heraclius. He said Muhammad (ﷺ) had never been accused of lying before he claimed prophethood.
He refused to compromise the message or change the Qur'an when his opponents asked.
He died with his armour pawned for thirty measures of barley.
His own sayings were kept apart from the Qur'an, while the Bible includes its writers' personal greetings.
The Islamic perspective
The Qur'an is the word of God (﷾), brought by the angel Gabriel to the Prophet (ﷺ). He recited it exactly as he received it and added nothing of his own.
The Qur'an says the Prophet (ﷺ) never read or wrote a book, and calls him the unlettered Prophet (ﷺ).
At the first revelation he told the angel three times that he could not read.
The Qur'an speaks as God (﷾) in the first person. It uses the command "Say" hundreds of times, most of them addressed to the Prophet (ﷺ).
The Qur'an rebukes the Prophet (ﷺ) in words he then had to recite in public.
The Qur'an tells details of earlier prophets, such as Noah's drowned son, that are in no earlier scripture.
The Qur'an offers its own freedom from contradiction as proof that it comes from God (﷾).