Short answer
No. In John 14:6 Jesus calls himself the way to the Father. A way leads to a place, and that place is God. Four verses earlier he tells his disciples that his Father's house has many mansions, and that he goes to prepare a place for them. In Matthew he said he was sent only to the lost sheep of Israel. So he was the way for the people he was sent to, as earlier prophets were for theirs. He also warned that calling him "Lord, Lord" saves no one. Only the one who does the will of the Father will enter. The Qur'an says Jesus foretold a messenger to come after him. That messenger, the Prophet Muhammad (ﷺ), shows the same way now. In the Qur'an, Jesus says: Allah (﷾) is my Lord and your Lord, so worship Him.
Full explanation
What Christians take from the verse
Christians read John 14:6 to mean that no one, in any age or place, can reach God (﷾) except through faith in Jesus. On this reading, only those who believe in him are saved.
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
What the verse is about
Jesus is speaking to his disciples at the Last Supper, his final meal with them. He says he is going away. He says his Father's house has many mansions, which means many rooms. He goes to prepare a place there for them. Then Thomas, one of the twelve, asks how they can know the way. John 14:6 is Jesus' answer to Thomas. So the promise is made to the disciples in the room: a place prepared for them.
“In my Father’s house are many mansions: if it were not so, I would have told you. I go to prepare a place for you.”
“Thomas saith unto him, Lord, we know not whither thou goest; and how can we know the way?”
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
The way is not the destination
In the verse, the Father is where the way leads, and Jesus is the way to Him. Later in the same chapter Jesus says his Father is greater than he is. In his prayer he says eternal life is to know the Father as the only true God (﷾), and Jesus as the one He sent. A messenger shows the way to God (﷾). He is not God.
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
“Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.”
“And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
Sent to his own people, for his own time
The Qur'an says God (﷾) sent a messenger to every nation. Jesus said he was sent only to the lost sheep of the house of Israel, meaning the Israelites. The Qur'an records that he told the Children of Israel he was God's (﷾) messenger to them. It adds that he gave the good news of a messenger to come after him, named Ahmad. Muslims understand this as the Prophet Muhammad (ﷺ). The Qur'an also praises the Jews and Christians who believed in God (﷾) and the Last Day, the Day of Judgment, and did good. Muslim scholars explain that this means those who followed the messenger of their own time. Once the last messenger came, submitting to God (﷾) means following him.
“For We assuredly sent amongst every People a messenger, (with the Command), "Serve Allah, and eschew Evil": of the People were some whom Allah guided, and some on whom error became inevitably (established). So travel through the earth, and see what was the end of those who denied (the Truth).”
“But he answered and said, I am not sent but unto the lost sheep of the house of Israel.”
“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!"”
“Those who believe (in the Qur'an), and those who follow the Jewish (scriptures), and the Christians and the Sabians,- any who believe in Allah and the Last Day, and work righteousness, shall have their reward with their Lord; on them shall be no fear, nor shall they grieve.”
“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).”
Calling him Lord is not enough
Jesus said that not everyone who calls him "Lord, Lord" will enter the kingdom of heaven. Only the one who does the will of his Father will enter. He added that on that day many will say they worked wonders in his name, and he will send them away. So faith in his name alone is not the way he taught.
“Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.”
“Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works? And then will I profess unto them, I never knew you: depart from me, ye that work iniquity.”
No one is condemned for a message that never reached him
Many Christians take the verse to mean that some people are lost through no fault of their own. These are all who die without ever hearing of Jesus. The Qur'an says God (﷾) does not punish a people until He has sent them a messenger.
“Who receiveth guidance, receiveth it for his own benefit: who goeth astray doth so to his own loss: No bearer of burdens can bear the burden of another: nor would We visit with Our Wrath until We had sent a messenger (to give warning).”
The way Jesus taught, in the Qur'an
The Qur'an gives the way Jesus taught in his own words. He said that Allah (﷾) is his Lord and their Lord, so serve Him, and this is a straight way.
“Verily Allah is my Lord and your Lord: Him therefore serve ye: this is a Way that is straight.”
Verse by verse
John 14:6 — Gospel of John, chapter 14, verse 6
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
Cited for
No one comes to the Father except through Jesus. So only those who believe in him are saved, in every age and every land.
The response
Four verses earlier Jesus tells his disciples there are many mansions, and he goes to prepare a place for them. He answers Thomas about where he himself is going. He calls the Father greater than himself. And he said he was sent only to the lost sheep of Israel.
Where it lands — It makes the Father the destination and Jesus the way to Him for those he was sent to. It does not make him the only way for all peoples in all ages.
Matthew 28:19 — Gospel of Matthew, chapter 28, verse 19
“Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:”
Cited for
Christians say Jesus rose from the dead and then sent his disciples to all nations, so he is the way for everyone.
The response
During his years of teaching he said he was sent only to the lost sheep of Israel. He told the twelve not to go to the Gentiles, who are non-Jews. When he later sent them to the nations, it was to teach them "to observe all things whatsoever I have commanded you". He named the first commandment as the oneness of God (﷾). And he told a man who asked about eternal life to keep the commandments.
Where it lands — The disciples were sent to teach what he commanded. That is a way to God (﷾) through obedience, not a claim that he is the only way for every age.
John 10:16 — Gospel of John, chapter 10, verse 16
“And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd.”
Cited for
Jesus said he had other sheep not of this fold, so his way is for all peoples.
The response
In Matthew he said plainly that he was sent only to the lost sheep of Israel. He told the twelve not to go to the Gentiles. The "other sheep" are not named, so they cannot outweigh his plain words about whom he was sent to.
Where it lands — The verse names no one, while Matthew names Israel as the people he was sent to.
Perspectives
Mainstream Christian
Jesus is the only way to God for all people in every age. Salvation comes through faith in him alone.
In John 14:6 Jesus says no one comes to the Father except through him. Christians read this as meaning salvation is only by faith in Jesus.
Rebuttal
John records these words, but Jesus spoke them to his disciples, promising them a place in his Father's house. He calls the Father greater than himself, and says he was sent only to the lost sheep of Israel. And he warned that calling him "Lord, Lord" saves no one who does not do the Father's will.
The Islamic perspective
Jesus was the way to God (﷾) for the people he was sent to, in his time. The way he taught was to worship the One God (﷾) and do His will. After him God sent the Prophet Muhammad (ﷺ), and following him is now the way to God (﷾).
Just before John 14:6, Jesus speaks of a house of many mansions. He tells his disciples he goes to prepare a place for them there.
The promise of "a place" is made to his own disciples. That does not make him the only way for every people in every age.
In the verse the Father is the destination and Jesus is the way to Him. And Jesus says the Father is greater than he is.
Jesus said he was sent only to the lost sheep of the house of Israel.
The Qur'an says God (﷾) sent a messenger to every nation.
Jesus said calling him Lord will not save anyone who does not do the will of the Father.
The Qur'an says Jesus foretold a messenger after him named Ahmad.
The Qur'an teaches that God (﷾) does not punish a people until a messenger has come to them.
The Qur'an praises believing Jews and Christians. Muslim scholars read this as praise for those who followed the messenger of their own time.