The Satanic Verses incident refers to an event in which Muhammad allegedly recited verses given to him by Satan, believing that they were verses of the Quran given by Gabriel.
An account of this event is given in The History of al-Tabari, which was written by the same author that produced one of Islam's most important Quranic commentaries, Muhammad al-Tabari (923 AD):
The History of al-Tabari, Volume 6, pg. 108-110
When the Messenger of God saw how his tribe turned their backs on him and was grieved to see them shunning the message he had brought to them from God, he longed in his soul that something would come to him from God which would reconcile him with his tribe. With his love for his tribe and his eagerness for their welfare it would have delighted him if some of the difficulties which they made for him could have been smoothed out, and he debated with himself and fervently desired such an outcome.
Then God revealed: By the Star when it sets, your comrade does not err, nor is he deceived; nor does he speak out of (his own) desire ... and when he came to the words: Have you thought upon al-Lat and al-'Uzza and Manat, the third, the other? Satan cast on his tongue, because of his inner debates and what he desired to bring to his people, the words: These are the high-flying cranes; verily their intercession is accepted with approval.
When Quraysh heard this, they rejoiced and were happy and delighted at the way in which he spoke of their gods, and they listened to him, while the Muslims, having complete trust in their Prophet in respect of the messages which he brought from God, did not suspect him of error, illusion, or mistake. When he came to the prostration, having completed the surah, he prostrated himself and the Muslims did likewise, following their Prophet, trusting in the message which he had brought and following his example.
Those polytheists of the Quraysh and others who were in the mosque likewise prostrated themselves because of the reference to their gods which they had heard, so that there was no one in the mosque, believer or unbeliever, who did not prostrate himself. The one exception was al-Walid b. al-Mughirah, who was a very old man and could not prostrate himself; but he took a handful of soil from the valley in his hand and bowed over that. Then they all dispersed from the mosque.
The Quraysh left delighted by the mention of their gods which they had heard, saying, "Muhammad has mentioned our gods in the most favorable way possible, stating in his recitation that they are the high-flying cranes and that their intercession is received with approval." The news of this prostration reached those of the Messenger of God's Companions who were in Abyssinia and people said, "The Quraysh have accepted Islam." Some rose up to return, while others remained behind.
Then Gabriel came to the Messenger of God and said, "Muhammad, what have you done? You have recited to the people that which I did not bring to you from God, and you have said that which was not said to you." Then the Messenger of God was much grieved and feared God greatly, but God sent down a revelation to him, for He was merciful to him, consoling him and making the matter light for him, informing him that there had never been a prophet or a messenger before him who desired as he desired and wished as he wished but that Satan had cast words into his recitation, as he had cast words on Muhammad's tongue. Then God cancelled what Satan had thus cast, and established his verses by telling him that he was like other prophets and messengers, and revealed:
"Never did we send a messenger or a prophet before you but that when he recited (the Message) Satan cast words into his recitation (umniyyah). God abrogates what Satan casts. Then God established his verses. God is knower, wise."
Thus God removed the sorrow from his Messenger, reassured him about that which he had feared and cancelled the words which Satan had cast on his tongue, that their gods were the highflying cranes whose intercession was accepted with approval. He now revealed, following the mention of "al-Lat, al-'Uzza and Manat, the third, the other," the words: Are yours the males and his the females? That indeed were an unfair division! They are but names which you have named, you and your fathers ... to the words: to whom he wills and accepts.' This means, how can the intercession of their gods avail with God?
When Muhammad brought a revelation from God cancelling what Satan had cast on the tongue of His Prophet, the Quraysh said, "Muhammad has repented of what he said concerning the position of your gods with God, and has altered it and brought something else." Those two phrases which Satan had cast on the tongue of the Messenger of God were in the mouth of every polytheists, and they became even more ill-disposed and more violent in their persecution of those of them who had accepted Islam and followed the Messenger of God.
The account is repeated again in Volume 6, in which it is recorded that Muhammad said the following when Gabriel first informed him that he had recited from Satan:
The History of al-Tabari, Volume 6, pg. 111
That evening Gabriel came to him and reviewed the surah with him, and when he reached the two phrases which Satan had cast upon his tongue he said, "I did not bring you these two." Then the Messenger of God said, "I have fabricated things against God, and have imputed to Him words which He has not spoken."
In reviewing the whole account, notice that:
A nearly identical account to the one from al-Tabari above is also given in Sirat Rasul Allah ("The Life of the Prophet"), which is the earliest biography of Muhammad, written by Ibn Ishaq (704-767 AD) and Ibn Hisham (833 AD), under the section entitled "His Temporary Concession to Polytheism". In total, there are dozens of Muslim sources from the first few centuries of Islam that reference the event in some form - Shahab Ahmed catalogues 50 in his book Before Orthodoxy.
The Satanic Verses incident, if it did indeed occur, provides the background of two verses of the Quran. The first passage is the one which originally contained the Satanic verses, from Surah 53, which now reads:
Surah 53:19-22
19 Now, have you considered ˹the idols of˺ Lat and Uzza,
20 and the third one, Manat, as well?
21 Do you ˹prefer to˺ have sons while ˹you attribute˺ to Him daughters?
22 Then this is ˹truly˺ a biased distribution!
The second passage is the one which Gabriel used to comfort Muhammad, in which he tells Muhammad that Satan caused all the prophets to do something similar to what Muhammad had just done:
Surah 22:52
52 Never did We send a Messenger or a Prophet before you, but; when he did recite the revelation or narrated or spoke, Shaitan (Satan) threw (some falsehood) in it. But Allah abolishes that which Shaitan (Satan) throws in. Then Allah establishes His Revelations. And Allah is All-Knower, All-Wise:
Muslims, in response, disregard the Satanic Verses incident as a fabrication. However, this runs contrary to the very earliest source on the life of Muhammad, the biography of Ibn Ishaq. It also makes little sense from a logical point of view, as it proposes that early Muslims invented a story of their own prophet being deceived by Satan, and reciting false revelation, which seems extremely unlikely.
If the Satanic Verses incident took place, it disqualifies Muhammad completely from being in the line of the Biblical prophets, as the Bible says:
Deuteronomy 18:20
20 But the prophet, which shall presume to speak a word in my name, which I have not commanded him to speak, or that shall speak in the name of other gods, even that prophet shall die.
Muhammad, had he been alive at the time of Moses and Joshua, would have been put to death for the Satanic Verses incident, on the basis of his own testimony.
Additionally, the verse in the Quran that states "Never did We send a Messenger or a Prophet before you, but; when he did recite the revelation or narrated or spoke, Shaitan (Satan) threw (some falsehood) in it", is directly contradictory to this verse, if understood the sense of an incident like the Satanic Verses happening to all the prophets.
Adding to a litany of other difficulties, Islam also depends on early Muslims fabricating a narrative about their own prophet reciting verses from Satan. Otherwise, if the account is not dismissed as fabricated, Muhammad is immediately disqualified from Biblical prophethood.