To whom does the phrase "House of Israel" refer?
From Matthew 10:5-6 we see that the people populating "into the way of the Gentiles" or "a city of Samaritans" were not part of the House of Israel. Thus the claim that the house refers to Gentiles is bogus. The lost sheep (not reconciled to God) Jews, blood line descendants of Israel, the twelve tribes, are those to whom Christ first declared the gospel. His disciples had been looking for the Messiah. Thus at least some were open to His gospel, a fruitful initial ministry and fulfillment of prophesy. God will send a shepherd for lost sheep among His chosen people, referring to blood line descendants who were led astray becoming non or errant believers.
Jesus was not sent only among the lost sheep of Israel, as demonstrated by the Canaanite women.
Jesus Christ came to seek and to save that which was lost. Gentiles were never the possessions of God and so they cannot be lost.Jesus was not seeking the woman in Mt 15. It was a chance meeting as far as a meeting can be a chance with God. The woman came to him, he did not go to her. He did not address her as a sheep but he did categorize her as a dog. The gentiles will be an object of his concern at some point in his post resurrection ministry and he clearly says that in this passage when he deals with other gentiles.
John 12:20 And there were certain Greeks among them that came up to worship at the feast:
21 The same came therefore to Philip, which was of Bethsaida of Galilee, and desired him, saying, Sir, we would see Jesus.
22 Philip cometh and telleth Andrew: and again Andrew and Philip tell Jesus.
23 And Jesus answered them, saying, The hour is come, that the Son of man should be glorified.
24 Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit.
25 He that loveth his life shall lose it; and he that hateth his life in this world shall keep it unto life eternal.
26 If any man serve me, let him follow me; and where I am, there shall also my servant be: if any man serve me, him will my Father honour.
27 Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour.
28 Father, glorify thy name. Then came there a voice from heaven, saying, I have both glorified it, and will glorify it again.
29 The people therefore, that stood by, and heard it, said that it thundered: others said, An angel spake to him.
30 Jesus answered and said, This voice came not because of me, but for your sakes.
31 Now is the judgment of this world: now shall the prince of this world be cast out.
32 And I, if I be lifted up from the earth, will draw all men unto me.
33 This he said, signifying what death he should die.
34 The people answered him, We have heard out of the law that Christ abideth for ever: and how sayest thou, The Son of man must be lifted up? who is this Son of man?
35 Then Jesus said unto them, Yet a little while is the light with you. Walk while ye have the light, lest darkness come upon you: for he that walketh in darkness knoweth not whither he goeth.
36 While ye have light, believe in the light, that ye may be the children of light. These things spake Jesus, and departed, and did hide himself from them.
37 But though he had done so many miracles before them, yet they believed not on him:
38 That the saying of Esaias the prophet might be fulfilled, which he spake, Lord, who hath believed our report? and to whom hath the arm of the Lord been revealed?
39 Therefore they could not believe, because that Esaias said again,
40 He hath blinded their eyes, and hardened their heart; that they should not see with their eyes, nor understand with their heart, and be converted, and I should heal them.
41 These things said Esaias, when he saw his glory, and spake of him.
42 Nevertheless among the chief rulers also many believed on him; but because of the Pharisees they did not confess him, lest they should be put out of the synagogue:
43 For they loved the praise of men more than the praise of God.
The fruit of his death, burial, and resurrection will be the gentiles since the Jews failed to believe as the scriptures say to those who have eyes to see and do not attempt to re-write the scriptures to suit themselves.
Check it out. Paul the apostle to the gentiles wrote 13 letters and never one time said we were lost. Gentiles are not lost, they are alienated from God by sin. The death and resurrection achieve reconciliation on both sides of the issue because Jesus Christ is the one mediator between God and man.
Believe the words.