I think it may be helpful to comment on the two seeds of Abraham; for there are two seeds with two different promises.
Firstly, there is a physical seed to whom are given physical promises- a great nation and a physical land for it to dwell in. These promises were received by Israel in full (Josh 21:43-45). This seed ‘after the flesh’ (Gal 4:29) is represented by Ishmael. It is most important to understand that Ishmael is not in the Covenant. ‘And Abraham said to God, “Oh, that Ishmael might live before you!” Then God said, “No, Sarah your wife shall bear you a son and you shall call his name Isaac. I will establish My covenant with him”’ (Gen 17:18-19). Nothing could be clearer than this; Ishmael is not in the covenant, although he receives the covenant sign (v26). Ishmael, though not an Israelite, is a type of Israel after the flesh. He receives the earthly promises (Gen 17:20) and the outward sign, but not the spiritual blessings (Gal 4:30; Acts 7:51-53). He persecutes the True Seed (Gen 21:9; John 8:37ff; Gal 4:29). His circumcision is of no avail to him since he lacks what circumcision symbolized; a humble, circumcised heart (Jer 9:25-26).
There is also a spiritual seed of Abraham; those who are in Christ, the True Seed, by faith. These are they who are looking for a heavenly country just as Abraham was. Just as Abraham did not receive an earthly inheritance (Acts 7:5 etc), so the true Israelite knew that Canaan was not his true home (Psalm 39:12; 119:19. cf. 1Peter 2:11). He put no confidence in his circumcision, but rather his circumcision spoke to him of the promised Seed of Abraham who should come (cf. Luke 2:25-32; Phil 3:3).
It may be asked then, are there two Abrahamic covenants, one to the physical descendants of Abraham and one to his spiritual seed? No, for as we have seen, Ishmael receives certain promises, and is given the sign of circumcision, but he is not in the covenant. But what then of Gen 17:10ff, which says, “This is My covenant which you shall keep between me and you and your descendants (lit. ‘Seed’) after you. Every male child among you shall be circumcised; and you shall be circumcised in the flesh of your foreskins and it shall be a sign of the covenant between me and you………and the uncircumcised male child who is not circumcised in the flesh of his foreskin, that person shall be cut off from his people; he has broken My covenant”? How could someone break the covenant if he’s not in it?
As so often in the Scriptures, we can use the New Testament to shed light upon the Old. First, we can look at circumcision: ‘And [Abraham] received the sign of circumcision, a seal of the righteousness of the faith which he had while still uncircumcised, that he might be the father of those who believe, though they are uncircumcised, that righteousness might be imputed to them also’ Rom 4:11). First we should note that circumcision was not a seal of anything to anybody but Abraham, and even to him it was simply a confirmation of the blessings that had been already promised to him. It was a divine pledge to him that from him should come that Seed through whom all nations should be blessed. It was not a seal of his faith, but of the righteousness that should, in due time, be wrought by Christ in Whom he had believed (cf. John 8:56).
What then did circumcision signify to Abraham’s physical male progeny and to his male servants? Well, firstly it had nothing to do with faith. It was a requirement for service in Abraham’s household. If you wanted to work for Mr Abraham, you had the snip. Indeed, nowhere in the entire Bible will you find physical circumcision connected with anybody’s faith but Abraham’s. In itself it signified precisely nothing. From the descendants of Abraham, the Messiah would be born, but it did not follow that any particular descendant should be an ancestor of Christ. Circumcision was a sign, not to Abraham’s physical seed, but to those ‘Israelites indeed’ (John 1:47) who were looking forward to the Christ by faith, that the promises of God should eventually be fulfilled.
Next we can look at the ‘promises:’ ‘Now to Abraham and his Seed were the promises made. He does not say, “And to seeds,” as of many, but as of one, “And to your Seed,” who is Christ……..and if you are Christ’s then you are Abraham’s seed, and heirs according to the promise’ (Gal 3:16, 29). In the light of this divinely-inspired commentary on Gen 17, it is perverse of the NKJV and other modern translations to speak of ‘descendants’ in Gen 17:10 and elsewhere, instead of ‘seed.’ The Authorised Version is more reliable at this point. The spiritual promises of the Abrahamic Covenant never applied to those who were physical descendants of Abraham, but to those of all nations (including Israel, of course) who are in Christ by faith. Very solemn are the words of our Lord on this matter: “And I say to you that many will come from east and west and will sit down with Abraham, Isaac and Jacob in the kingdom of heaven, but the sons of the kingdom will be cast into outer darkness. There will be weeping and gnashing of teeth” (Matt 8:11-12).
So we see that the Abrahamic Covenant is tied up with the coming of the promised son- the miracle child (Gen 15:2-6). Isaac is not Christ, but he is a type or foreshadowing of Christ: long promised, born miraculously, persecuted by his own kin (Ishmael), offered up by his father, who received him (figuratively- Heb 11:19) back from the dead. The children of God come from him (Rom 9:7; Heb 3:5b). It is worth reading Isaiah 54 prayerfully in the light of these points.
So are Christians in some way ‘under’ the Abrahamic Covenant? The reader may search the whole Bible through but he will find no indication that they are. Believers are the true children of Abraham, and we read in Gal 3:14 that, ‘…….. the blessing of Abraham might come upon the Gentiles in Christ Jesus, that we might receive the promise of the Spirit through faith.’ There is no promise here for the children of believers, any more than there was for Abraham’s children by Hagar or Keturah (Rom 9:7 again!). ‘Know therefore that only those who are of faith are sons of Abraham’ (Gal 3:7).
In the words of A.W.Pink: “The grand design of God’s covenant with Abraham was to make known that through him should come the One who would bring blessing to all the families of the earth.” But in order for this design to come to pass, it was necessary for a nation to arise for Christ to be born into, so that His earthly genealogy might be preserved. Pink continues, “Abraham is called a ‘father’ neither in a federal nor in a spiritual sense, but because he is the head of the faith clan, the prototype to which all believers are conformed. Christians are not under the Abrahamic covenant, though they are ‘blessed with him’ by having their faith counted unto righteousness. Though New Testament believers are not under the Abrahamic covenant, they are, because of their union with Christ, heirs of its spiritual inheritance.” [A.W. Pink, The Divine Covenants]