Short answer
Christians should follow the Prophet Muhammad (ﷺ) because God (﷾) Himself commands it in the Torah. He promised Moses a prophet like him, with God's (﷾) words in his mouth. Then He warned that whoever will not listen to the words this prophet speaks in His name, He will require it of him. The Qur'an is spoken in God's (﷾) name. The first words revealed to him told him to read in the name of his Lord. And 113 of its 114 chapters open "In the name of Allah". Jesus said he had many things still to say that his followers could not yet bear. He also told the chief priests and elders that the kingdom of God (﷾) would be taken from them and given to another nation. If the Bible foretells Muhammad (ﷺ), turning away from him means turning away from the Bible. Following him does not mean leaving Jesus.
Full explanation
Why Christians see no need
Christians hold that salvation is complete in Jesus. Peter, one of Jesus' disciples, told the Jewish council that there is no other name under heaven by which people must be saved. The letter to the Hebrews, a book of the New Testament, says that God (﷾) spoke through the prophets in times past. It says that in these last days He has spoken through His Son. On this view, nothing that comes later can add to what Jesus did.
“Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.”
“God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;”
The warning in the Torah
In the Torah (the first five books of the Bible), God (﷾) promised Moses a prophet like him. This prophet would come from among Israel's brethren (their kinsmen), with God's (﷾) words in his mouth. In the very next verse God (﷾) gave a warning. He would require it of anyone who would not listen to the words that prophet speaks in His name. The warning has no exception for those who follow an earlier prophet. The Torah adds that no prophet like Moses arose again in Israel. Muhammad (ﷺ) matches Moses point after point. That comparison is set out in full on its own page.
“I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth; and he shall speak unto them all that I shall command him.”
“And it shall come to pass, that whosoever will not hearken unto my words which he shall speak in my name, I will require it of him.”
“And there arose not a prophet since in Israel like unto Moses, whom the Lord knew face to face,”
"That prophet" was still awaited
The Gospel of John tells how priests sent from Jerusalem asked John the Baptist who he was. He said he was not the Christ. They then asked whether he was Elijah, and whether he was "that prophet". The people of Jesus' own day were waiting for a prophet besides the Christ.
“And this is the record of John, when the Jews sent priests and Levites from Jerusalem to ask him, Who art thou? And he confessed, and denied not; but confessed, I am not the Christ. And they asked him, What then? Art thou Elias? And he saith, I am not. Art thou that prophet? And he answered, No.”
Words spoken in God's (﷾) name
Muhammad (ﷺ) never claimed the Qur'an as his own words. The first words revealed to him were a command to read in the name of his Lord. Every chapter of the Qur'an but one opens with "In the name of Allah, Most Gracious, Most Merciful", and Muslims begin their recitation with it. The Qur'an says he does not speak from his own desire. It is revelation sent down to him. That is the prophet Deuteronomy describes, speaking God's (﷾) words in God's (﷾) name.
“Proclaim! (or read!) in the name of thy Lord and Cherisher, Who created- Created man, out of a (mere) clot of congealed blood: Proclaim! And thy Lord is Most Bountiful,- He Who taught (the use of) the pen,- Taught man that which he knew not.”
“In the name of Allah, Most Gracious, Most Merciful.”
“Nor does he say (aught) of (his own) Desire. It is no less than inspiration sent down to him:”
The Qur'an repeats the warning
Deuteronomy says God (﷾) will require it of those who will not listen. The Qur'an confirms the same warning in its own words. A religion other than submission to God (﷾) will not be accepted, and in the Hereafter the one who seeks it will be among the losers.
“If anyone desires a religion other than Islam (submission to Allah), never will it be accepted of him; and in the Hereafter He will be in the ranks of those who have lost (All spiritual good).”
“And it shall come to pass, that whosoever will not hearken unto my words which he shall speak in my name, I will require it of him.”
Jesus said there was more to come
Jesus did not present himself as the last word. He told his disciples that he had many things still to say that they could not bear yet. He said that one would come who would guide them into all truth, speaking only what he hears. Jesus calls him the Spirit of truth. That title fits a prophet, not the Holy Spirit of Christian doctrine. A separate page on the Paraclete (the Comforter, or Helper, Jesus promised) explains why. The Qur'an records that Jesus gave glad tidings of a messenger to come after him, named Ahmad. Ahmad is a name of the Prophet (ﷺ) from the same root as Muhammad.
“I have yet many things to say unto you, but ye cannot bear them now. Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: and he will shew you things to come. He shall glorify me: for he shall receive of mine, and shall shew it unto you.”
“And remember, Jesus, the son of Mary, said: "O Children of Israel! I am the messenger of Allah (sent) to you, confirming the Law (which came) before me, and giving Glad Tidings of a Messenger to come after me, whose name shall be Ahmad." But when he came to them with Clear Signs, they said, "this is evident sorcery!"”
The kingdom given to another nation
Jesus cursed a fig tree that bore no fruit. He also told a parable of a fig tree given one last year before it was cut down. Then he told his hearers that the kingdom of God (﷾) would be taken from them and given to a nation bringing forth its fruits. He said the stone the builders rejected would become the head of the corner. God (﷾) had promised Abraham that He would make a nation of Ishmael too. Six centuries after Jesus came the Prophet Muhammad (ﷺ), a descendant of Ishmael. He described himself as the one missing brick in the corner of the house the prophets had built. He also said he was the last of the prophets.
“And when he saw a fig tree in the way, he came to it, and found nothing thereon, but leaves only, and said unto it, Let no fruit grow on thee henceforward for ever. And presently the fig tree withered away.”
“He spake also this parable; A certain man had a fig tree planted in his vineyard; and he came and sought fruit thereon, and found none. Then said he unto the dresser of his vineyard, Behold, these three years I come seeking fruit on this fig tree, and find none: cut it down; why cumbereth it the ground? And he answering said unto him, Lord, let it alone this year also, till I shall dig about it, and dung it: And if it bear fruit, well: and if not, then after that thou shalt cut it down.”
“Jesus saith unto them, Did ye never read in the scriptures, The stone which the builders rejected, the same is become the head of the corner: this is the Lord’s doing, and it is marvellous in our eyes? Therefore say I unto you, The kingdom of God shall be taken from you, and given to a nation bringing forth the fruits thereof.”
“And whosoever shall fall on this stone shall be broken: but on whomsoever it shall fall, it will grind him to powder.”
“And also of the son of the bondwoman will I make a nation, because he is thy seed.”
Abu Hurayra reports that the Prophet (ﷺ) compared himself and the prophets before him to a man who built a fine house but left the place of one brick empty in a corner. People walked round it, admired it, and wished the brick were in place. He said: I am that brick, and I am the last of the prophets.
Following him does not mean leaving Jesus
A Muslim believes in Jesus as one of God's (﷾) mighty messengers. The Qur'an commands belief in all the messengers without making a distinction between them. It also commands belief in what was given to Moses and Jesus. So a Christian who follows Muhammad (ﷺ) keeps Jesus. He also obeys what the Torah and Jesus both said was coming.
“The Messenger believeth in what hath been revealed to him from his Lord, as do the men of faith. Each one (of them) believeth in Allah, His angels, His books, and His messengers. "We make no distinction (they say) between one and another of His messengers." And they say: "We hear, and we obey: (We seek) Thy forgiveness, our Lord, and to Thee is the end of all journeys."”
“Say ye: "We believe in Allah, and the revelation given to us, and to Abraham, Isma'il, Isaac, Jacob, and the Tribes, and that given to Moses and Jesus, and that given to (all) prophets from their Lord: We make no difference between one and another of them: And we bow to Allah (in Islam)."”
Verse by verse
Hebrews 1:1-2 — Epistle to the Hebrews, chapter 1, verses 1-2
“God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;”
Cited for
God's last word was spoken through His Son. After Jesus there is nothing more to hear.
The response
These are the words of a letter, not the words of Jesus. Jesus himself said he had many things still to say that his disciples could not bear. He said that one was coming to guide them into all truth. The case that this guide is the Prophet (ﷺ), and not the Holy Spirit, is on the Paraclete page. And the warning of Deuteronomy 18:19 has no exception for those who follow an earlier prophet.
Where it lands — Jesus' own words leave the door open that the letter closes.
Acts 3:22 — Acts of the Apostles, chapter 3, verse 22
“For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you.”
Cited for
Peter says the prophet like Moses is Jesus, so Christians have already listened to him.
The response
John the Baptist told the priests of Jesus' day that he was not the Christ. As a separate question, they then asked whether he was "that prophet". They kept the two apart. Moses' own summary in Deuteronomy 18:15 says "from the midst of thee, of thy brethren". But God's (﷾) words in 18:18 say only "from among their brethren". Deuteronomy calls the Edomites, the sons of Esau, Israel's "brethren" (2:4). The sons of Ishmael are brethren in the same sense.
Where it lands — The witnesses in John's Gospel themselves kept "that prophet" apart from the Christ. The Torah uses "brethren" for Israel's kin as well as for Israel. The full case, including Deuteronomy 17:15, is on the prophet-like-unto-Moses page.
Perspectives
Mainstream Christian
Christians hold that Jesus' death and resurrection completed salvation. God's final word, they say, was spoken through His Son, so no later prophet is needed.
Peter said there is no other name under heaven by which people must be saved.
Rebuttal
Peter did say this. But Jesus told the Father that eternal life is to know "the only true God, and Jesus Christ, whom thou hast sent" (John 17:3). Islam teaches exactly that, and Deuteronomy 18:19 warns anyone who will not listen to the prophet who speaks in God's (﷾) name.
Hebrews says God has spoken in these last days by His Son.
Rebuttal
The letter says God (﷾) has spoken by His Son in these last days, but it does not say He would never speak again. Jesus himself said he had many things still to say that his disciples could not bear. And Deuteronomy 18:19 has no exception for those who follow an earlier prophet.
Peter applied the promise of a prophet like Moses to Jesus.
Rebuttal
Peter did apply it to Jesus. But John the Baptist told the priests he was not the Christ, and they then asked separately whether he was "that prophet". Deuteronomy also calls the Edomites Israel's "brethren" (2:4), so the sons of Ishmael are brethren in the same sense. The full case, with Deuteronomy 18:15, is on the page on the prophet like Moses.
The Islamic perspective
God (﷾) warned in the Torah that He would require it of anyone who ignores the prophet who speaks His words in His name. Muhammad (ﷺ) is that prophet, and the Bible's own promises point to him.
Deuteronomy 18:19 gives a warning. God (﷾) will require it of anyone who will not listen to the prophet who speaks in His name.
The Qur'an's first revealed words command reading in the name of the Lord. And 113 of its 114 chapters open in the name of Allah (﷾).
The Qur'an is revelation, not the Prophet's (ﷺ) own words.
Jesus said he had many more things to say that his disciples could not yet bear.
Jesus told the chief priests and elders that the kingdom of God (﷾) would be taken from them. It would be given to another nation.
The prophet like Moses in Deuteronomy 18 is Muhammad (ﷺ), from Israel's brethren the Ishmaelites.
The Qur'an says no religion other than submission to God (﷾) will be accepted.