Did Jesus come to minister to the Jews only?

In Matthew 15, there is an incident that, at first, seems to confirm the idea that Jesus came only for the Jews. Jesus was traveling through Tyre and Sidon, a Gentile region, and “a Canaanite woman from that vicinity came to him, crying out, ‘Lord, Son of David, have mercy on me! My daughter is demon-possessed and suffering terribly’” (Matthew 15:22). This Gentile woman recognized Jesus as the Messiah (“Son of David”), but “Jesus did not answer a word” (verse 23). As the woman kept up her appeals, Jesus finally responded, but His words seemed to hold little hope: “I was sent only to the lost sheep of Israel” (verse 24). However, the woman did not give up, and Jesus eventually granted her request, based on her “great faith” (verse 28).

Tyre & Sidon was a Gentile city, so it is obvious he did not go there to minister to Jews only.

The fact that Jesus helped the Canaanite woman, even though His mission was to the Jews, is a significant detail in the Gospel narrative.

Throughout His earthly ministry, Jesus gave other indications that His power and compassion reached to all people.
He healed a Roman centurion’s servant (Luke 7:1–10).
He traveled through the Gentile region of the Gerasenes (Mark 5:1).
He ministered in a Samaritan city (John 4).

Jesus came to save everybody (1 John 2:2). Jesus Christ is God Himself (John 1:1). Jesus died on the cross as the payment for all our sins, and He rose from death in resurrection (1 Corinthians 15:3–4). Jesus said He was the Good Shepherd, and He predicted that His flock would be greatly expanded: “I have other sheep that are not of this sheep pen. I must bring them also. They too will listen to my voice, and there shall be one flock and one shepherd” (John 10:16).

It took a while for the early church to recognize that salvation was available to the Gentiles. The Jewish Christians who fled the persecution in Jerusalem went into the Gentile regions of Phoenicia, Cyprus and Antioch, but they were “spreading the word only among Jews” (Acts 11:19). Peter was hesitant to bring the gospel to a Gentile household, but God made it plain that Cornelius was also one of the elect (Acts 10).

“Is God the God of Jews only? Is he not the God of Gentiles too? Yes, of Gentiles too” (Romans 3:29). Jesus was the Jewish Messiah, but He had come to offer salvation to everybody. The Messiah was to be a “light for the Gentiles” (Isaiah 42:6). So call on Jesus, because “everyone who calls on the name of the Lord will be saved” (Acts 2:21).
 

Very Early in his ministry, large crowds followed him from Decapolis, a group of ten Greek cities.

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Matthew 4:25

Large crowds from Galilee, the Decapolis, Jerusalem, Judea and the region across the Jordan followed him.

In John Chapter 3, Jesus teaches Nicodemus, a member of the Jewish ruling council and emphasizes that His mission is for everyone, for the whole world.

Verse 14, "Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up,[f] 15 that everyone who believes may have eternal life in him."
16 "For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. 17 For God did not send his Son into the world to condemn the world, but to save the world through him."

Conclusion: Jesus maintained from his early ministry and all throughout it that he came to minister to the whole world.