Did Jesus Heal and Preach to Only Jews? No!
“I will put my Spirit upon him, and he shall proclaim justice to the Gentiles...”
Some misguided biblical skeptics contend that “the Jesus of Matthew” is utterly unconcerned with non-Jews (Gentiles). This is a ludicrous opinion. A clear instance in Matthew of Jesus’ outreach beyond the Jews is His interaction with the Roman centurion:
Matthew 8:5-13 (RSV) As he entered Caper’na-um, a centurion came forward to him, beseeching him  and saying, “Lord, my servant is lying paralyzed at home, in terrible distress.”  And he said to him, “I will come and heal him.”  But the centurion answered him, “Lord, I am not worthy to have you come under my roof; but only say the word, and my servant will be healed.  For I am a man under authority, with soldiers under me; and I say to one, `Go,’ and he goes, and to another, `Come,’ and he comes, and to my slave, `Do this,’ and he does it.”  When Jesus heard him, he marveled, and said to those who followed him, “Truly, I say to you, not even in Israel have I found such faith.  I tell you, many will come from east and west and sit at table with Abraham, Isaac, and Jacob in the kingdom of heaven,  while the sons of the kingdom will be thrown into the outer darkness; there men will weep and gnash their teeth.”  And to the centurion Jesus said, “Go; be it done for you as you have believed.” And the servant was healed at that very moment.
Note how Jesus not only readily healed the Roman centurion’s servant (8:7, 13), but also “marveled” at his faith and commended it as superior to the faith of anyone “in Israel” (8:10). And that led Him to observe that many Gentiles will be saved, whereas many Jews will not be saved (8:11-12). If this is supposedly a “Jewish only” view (“Gentiles need not apply”), it was surely the weirdest, most confusing way imaginable to express it.
A second counter-example is from the Sermon on the Mount. Jesus told His followers, “You are the light of the world” (Matthew 5:14).
A third example is the parable of the weeds (Matthew 13:37-38), which showed a universal mission field (“the world”) fifteen chapters before Matthew 28.
A fourth example is Jesus healing the demon-possessed daughter of the Syrophoenician woman (Matthew 15:21-28).
As a fifth example, Matthew seems to not be aware of his own supposed “Jews only Jesus” since he applies an Old Testament passage about outreach to Gentiles directly to Jesus as the Servant and Messiah:
Matthew 12:17-18, 20-21 This was to fulfil what was spoken by the prophet Isaiah:  “Behold, my servant whom I have chosen, my beloved with whom my soul is well pleased. I will put my Spirit upon him, and he shall proclaim justice to the Gentiles. . . .  he will not break a bruised reed or quench a smoldering wick, till he brings justice to victory;  and in his name will the Gentiles hope.”
A sixth counter-example is Jesus telling the Jewish “chief priests and scribes” (Matthew 21:15) and “Pharisees” (21:45) that righteous Gentiles will enter the kingdom before self-righteous Jews (like them):
Matthew 21:31b-32 “Truly, I say to you, the tax collectors and the harlots go into the kingdom of God before you.  For John came to you in the way of righteousness, and you did not believe him, but the tax collectors and the harlots believed him; and even when you saw it, you did not afterward repent and believe him.
Matthew 21:42-43, 45 Jesus said to them, “Have you never read in the scriptures: `The very stone which the builders rejected has become the head of the corner; this was the Lord’s doing, and it is marvelous in our eyes’?  Therefore I tell you, the kingdom of God will be taken away from you and given to a nation producing the fruits of it.”  When the chief priests and the Pharisees heard his parables, they perceived that he was speaking about them.
A seventh example is Jesus earlier echoing His message of the Great Commission (Matthew 28:19-20):
Matthew 24:14 And this gospel of the kingdom will be preached throughout the whole world, as a testimony to all nations; and then the end will come. (cf. Mark 13:10; Luke 24:47)
The same erroneous view is also supposedly apparent in the Gospel of Luke. Folks who argue in this fashion must not have read Jesus’ parable of the Good Samaritan (Luke 10:29-37). The whole point of it was to show that Samaritans were truly neighbors to Jews if they helped them, as the man did in the parable.
Secondly, Luke records Simeon saying about Jesus:
Luke 2:30-32 for mine eyes have seen thy salvation  which thou hast prepared in the presence of all peoples,  a light for revelation to the Gentiles, and for glory to thy people Israel.”
Thirdly, Jesus healed yet another foreigner: a Samaritan man, commending his faith (Luke 17:12-19).
Fourthly, Jesus specifically went to the land of the Gentile Gadarenes or Gerasenes, east of the Sea of Galilee, to minister to them. This was where Jesus sent the demons into the pigs (Jews, of course, kept no pigs), and it appears in all three synoptic Gospels (Mark 5:1-20; Luke 8:26-39; Matthew 8:28-34).
Jesus never says (nor does the entire New Testament ever say) that He came to “save Israel” or be the “savior of Israel.” He only claims to be the “Messiah” of Israel (John 4:25-26): which is a different thing. When Jesus says who it is that He came to save (i.e., provided they are willing), He states explicitly that He came “to save the lost” (Luke 19:10) and “to save the world” (John 12:47; cf. 1 Timothy 1:15).
Lastly, if we look at the Gospel of John, we observe Jesus’ interaction with the Samaritan woman at the well (John 4:5-29), which perfectly illustrates his “inclusive” view.