Short answer
Only to Israel, by his own words. In Matthew's Gospel, Jesus says he was sent only to "the lost sheep of the house of Israel". He told the twelve disciples not to go to the Gentiles (people who are not Jews) or to the Samaritans. Christians point to Matthew 28:19, where Jesus says to teach all nations. But years later, Peter still needed a vision before he would enter a Gentile's house. The believers in Jerusalem found fault with him for it. So even Peter, the leading disciple, did not act as if Jesus had sent him to the Gentiles. The Qur'an agrees that Jesus was a messenger to the Children of Israel. The Prophet Muhammad (ﷺ) said that every earlier prophet was sent to his own nation, but he was sent to all mankind.
Full explanation
What Jesus said about his mission
A woman of Canaan begged Jesus for help. He said he was not sent except to the lost sheep of the house of Israel. He said it was not right to give the children's bread to dogs. In the end he healed her daughter because of her faith. But he said plainly that his mission was not to her people. When he sent out the twelve, he told them not to go into the way of the Gentiles or into any city of the Samaritans. The Samaritans were a people living next to the Jews. He told the twelve they would not finish the cities of Israel before the Son of man (Jesus' name for himself) came. He also promised them twelve thrones, judging the twelve tribes of Israel.
“But he answered and said, I am not sent but unto the lost sheep of the house of Israel.”
“But he answered and said, It is not meet to take the children’s bread, and to cast it to dogs.”
“These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not: But go rather to the lost sheep of the house of Israel.”
“But when they persecute you in this city, flee ye into another: for verily I say unto you, Ye shall not have gone over the cities of Israel, till the Son of man be come.”
“And Jesus said unto them, Verily I say unto you, That ye which have followed me, in the regeneration when the Son of man shall sit in the throne of his glory, ye also shall sit upon twelve thrones, judging the twelve tribes of Israel.”
The first disciples kept to the Jews
Acts is the New Testament's history of the early church. In Acts, Peter tells a Gentile household that it is unlawful for a Jew to keep company with a man of another nation. He says he came only because God (﷾) showed him a vision. The believers in Jerusalem then found fault with him for eating with uncircumcised men. After Stephen was put to death, many believers were scattered. They travelled as far as Phoenicia, Cyprus and Antioch, preaching to none but Jews. Only some of them, later, spoke to Greeks at Antioch. Later an agreement left Peter, John, and James the brother of Jesus to go to the circumcision, meaning the Jews. Paul, a Christian missionary of the first century, went to the Gentiles. Paul never walked with Jesus, yet he made the Gentiles his mission.
“And he said unto them, Ye know how that it is an unlawful thing for a man that is a Jew to keep company, or come unto one of another nation; but God hath shewed me that I should not call any man common or unclean.”
“And when Peter was come up to Jerusalem, they that were of the circumcision contended with him, Saying, Thou wentest in to men uncircumcised, and didst eat with them.”
“Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews only.”
“And when James, Cephas, and John, who seemed to be pillars, perceived the grace that was given unto me, they gave to me and Barnabas the right hands of fellowship; that we should go unto the heathen, and they unto the circumcision.”
The verses Christians rely on
Christians point to the end of Matthew. There, in the Christian account of the resurrection, Jesus tells the eleven to teach all nations and baptise them. They add Mark 16:15, where he sends them into all the world to preach to every creature. They read the earlier limits as belonging only to the time before the resurrection. The answers to these verses are in the verse by verse section below.
“Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:”
“And he said unto them, Go ye into all the world, and preach the gospel to every creature.”
What the Qur'an says about Jesus' mission
The Qur'an calls Jesus a messenger to the Children of Israel. It records him telling them that he is the messenger of Allah (﷾) sent to them. His mission, like the mission of each prophet before him, was to one people.
“"And (appoint him) a messenger to the Children of Israel, (with this message): "'I have come to you, with a Sign from your Lord, in that I make for you out of clay, as it were, the figure of a bird, and breathe into it, and it becomes a bird by Allah's leave: And I heal those born blind, and the lepers, and I quicken the dead, by Allah's leave; and I declare to you what ye eat, and what ye store in your houses. Surely therein is a Sign for you if ye did believe;”
“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 prophet sent to all mankind
The Qur'an says the Prophet Muhammad (ﷺ) was sent to all men. It tells him to say that he is the messenger of Allah (﷾) to them all. It calls him a mercy for all creatures. He said that every earlier prophet was sent to his own nation only, but he was sent to all mankind. A prophet for one nation leaves room for one sent to everyone. The Qur'an also records Jesus giving glad tidings of a messenger to come after him.
“We have not sent thee but as a universal (Messenger) to men, giving them glad tidings, and warning them (against sin), but most men understand not.”
“Say: "O men! I am sent unto you all, as the Messenger of Allah, to Whom belongeth the dominion of the heavens and the earth: there is no god but He: it is He That giveth both life and death. So believe in Allah and His Messenger, the Unlettered Prophet, who believeth in Allah and His words: follow him that (so) ye may be guided."”
“We sent thee not, but as a Mercy for all creatures.”
Jabir ibn Abdullah reports that the Prophet (ﷺ) named five things given to him that no prophet before him was given. The last of them: every earlier prophet was sent to his own nation only, but he was sent to all mankind.
“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!"”
Verse by verse
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 that after the resurrection Jesus lifted the earlier limit and sent the disciples to all nations.
The response
Luke 24:47 and Acts 1:8 record similar words. Yet Acts shows Peter, the leading disciple, treating a visit to a Gentile as unlawful until a vision changed his mind. The church in Jerusalem finds fault with him for it. Many of the believers who are scattered travel as far as Antioch preaching to Jews only. Years later, Peter, John, and James the brother of Jesus still keep to the circumcision.
Where it lands — Peter needed a later vision before he would enter a Gentile's house, and Jerusalem found fault with him for it. That is not how men act on a standing command to make disciples of all nations. The verse also sits uneasily beside Matthew 10:5-6 and 15:24 in the same Gospel.
Mark 16:15 — Gospel of Mark, chapter 16, verse 15
“And he said unto them, Go ye into all the world, and preach the gospel to every creature.”
Cited for
Jesus sent the disciples into all the world to preach to every creature.
The response
This verse belongs to the longer ending of Mark, verses 9 to 20. Codex Sinaiticus and Codex Vaticanus both end Mark at verse 8, without it. A codex is an old handwritten book. These are the two oldest complete Greek manuscripts of Mark, both from the fourth century.
Where it lands — A verse missing from the oldest manuscripts cannot overturn what Jesus says twice in Matthew.
Acts 15:7 — Acts of the Apostles, chapter 15, verse 7
“And when there had been much disputing, Peter rose up, and said unto them, Men and brethren, ye know how that a good while ago God made choice among us, that the Gentiles by my mouth should hear the word of the gospel, and believe.”
Cited for
Peter says God chose him so that the Gentiles would hear the gospel from his mouth.
The response
Peter says God (﷾) made this choice, and Acts shows how. It came through a vision given to Peter after Jesus had gone. Peter does not say that Jesus had commanded it.
Where it lands — The move to the Gentiles came after Jesus, by a later vision, not by his own teaching.
Perspectives
Mainstream Christian
Christians say that during his ministry Jesus went to Israel first, and that after his resurrection he sent his disciples to every nation. They say the gospel is for all people.
In Matthew 28:19 Jesus tells the eleven to teach all nations.
Rebuttal
Matthew does record those words. But Peter, the leading disciple, still called it unlawful to visit a Gentile until a vision changed his mind. Years later, Peter, John and James still kept their mission to the Jews.
Christians read the limits in Matthew 10 and 15 as belonging to the time before the resurrection.
Rebuttal
Matthew 28:19 does speak of all nations. But Jesus had told the twelve they would not finish the cities of Israel before the Son of man came. And the believers scattered after Stephen's death preached to Jews only, while Peter needed a vision before he would visit a Gentile.
The Islamic perspective
Jesus was a mighty prophet sent to the Children of Israel, as he said himself and as the Qur'an confirms. The message for all mankind came with the Prophet Muhammad (ﷺ).
Jesus said he was not sent except to the lost sheep of the house of Israel.
Jesus told the twelve not to go to the Gentiles or to the Samaritans. They were to go to the lost sheep of Israel.
Peter needed a vision before he would visit a Gentile, and was criticised in Jerusalem for it. This shows that Peter did not act on a command to go to the Gentiles.
Many of the believers scattered after Stephen's death travelled as far as Antioch preaching to Jews only.
The Qur'an teaches that Jesus was a messenger to the Children of Israel.
The Qur'an teaches that the Prophet Muhammad (ﷺ) was sent to all mankind.
The Prophet Muhammad (ﷺ) said that earlier prophets were sent to their own nations, and he to all mankind.