When God made Abraham and his descendants the focus of his blessings, here is what was said in Genesis 22
17 That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore; and thy seed shall possess the gate of his enemies;
18 And in thy seed shall all the nations of the earth be blessed; because thou hast obeyed my voice.
Verse 18 is the key verse in understand how God related to Gentiles in the Old Testament, as well as the 4 Gospels.
Luke 7
2 And a certain centurion's servant, who was dear unto him, was sick, and ready to die.
3 And when he heard of Jesus, he sent unto him the elders of the Jews, beseeching him that he would come and heal his servant.
4 And when they came to Jesus, they besought him instantly, saying, That he was worthy for whom he should do this:
5 For he loveth our nation, and he hath built us a synagogue.
6 Then Jesus went with them. And when he was now not far from the house, the centurion sent friends to him, saying unto him, Lord, trouble not thyself: for I am not worthy that thou shouldest enter under my roof:
7 Wherefore neither thought I myself worthy to come unto thee: but say in a word, and my servant shall be healed.
8 For I also am a man set under authority, having under me soldiers, and I say unto one, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it.
9 When Jesus heard these things, he marvelled at him, and turned him about, and said unto the people that followed him, I say unto you, I have not found so great faith, no, not in Israel.
10 And they that were sent, returning to the house, found the servant whole that had been sick.
For the Roman Centurion case, the key verse was
3 And when he heard of Jesus, he sent unto him the elders of the Jews, beseeching him that he would come and heal his servant.
4 And when they came to Jesus, they besought him instantly, saying, That he was worthy for whom he should do this:
5 For he loveth our nation, and he hath built us a synagogue.
The Lord Jesus, without hesitation, immediately agreed to his request because he has blessed the Jews, and the Abrahamic covenant allows the blessing to then go to him.
As for the Canaanite woman, she did not deeds for the Jews but she wisely said in Mark
28 "Yes, Lord," she replied, "but even the dogs under the table eat the children's crumbs." 29 Then he told her, "
For such a reply, you may go; the demon has left your daughter."
Mark has this important detail which I highlighted in bold. That lady made the correct analogy that even the dogs, a term used by the Jews to describe Gentiles, could be blessed with the crumbs falling from the Jewish children's bread.
Thus Jesus healed her because her faith allowed her to be blessed, thru the Jews, as promised to Abraham.
