- Acts 3: 11While [the lame beggar who was just healed] clung to Peter and John, all the people ran together to them in the portico called Solomon's, astounded. 12And when Peter saw it he addressed the people: "Men of Israel, why do you wonder at this, or why do you stare at us, as though by our own power or piety we have made him walk? 13The God of Abraham, the God of Isaac, and the God of Jacob, the God of our fathers, glorified his servant Jesus, whom you delivered over and denied in the presence of Pilate, when he had decided to release him. 14But you denied the Holy and Righteous One, and asked for a murderer to be granted to you, 15and you killed the Author of life, whom God raised from the dead. To this we are witnesses. 16And his name--by faith in his name--has made this man strong whom you see and know, and the faith that is through Jesus has given the man this perfect health in the presence of you all.
17"And now, brothers, I know that you acted in ignorance, as did also your rulers. 18But what God foretold by the mouth of all the prophets, that his Christ would suffer, he thus fulfilled. 19Repent therefore, and turn again, that your sins may be blotted out, 20that times of refreshing may come from the presence of the Lord, and that he may send the Christ appointed for you, Jesus, 21whom heaven must receive until the time for restoring all the things about which God spoke by the mouth of his holy prophets long ago. 22Moses said, 'The Lord God will raise up for you a prophet like me from your brothers. You shall listen to him in whatever he tells you. 23And it shall be that every soul who does not listen to that prophet shall be destroyed from the people.' 24And all the prophets who have spoken, from Samuel and those who came after him, also proclaimed these days. 25You are the sons of the prophets and of the covenant that God made with your fathers, saying to Abraham, 'And in your offspring shall all the families of the earth be blessed.' 26God, having raised up his servant, sent him to you first, to bless you by turning every one of you from your wickedness." (ESV)
Let me make it clear before I put my foot in my mouth that I am not saying that children of believers are rightly reckoned as outside the new covenant as a point of fact. That's an overstatement, and it overlooks so many things that I would otherwise agree to from those making the claim in question about this passage.
However, this passage does not say that the New Covenant is handed down from paternal parent to child in a one-to-one correspondence. That's the chief flaw of the argument that this passage is being used to promote: the idea that we can presume that because parents are believers, children are rightly counted as believers in the fullest, soteriological sense of the word.
What Peter here preaches is that there is a heritage of faith. There's no question about that as he begins the sermon by establishing that God is the God of our fathers. And let's also make no doubt that these men Peter is preaching to are Jews, and therefore are rightly called "Israel" in the genealogical sense.
And in making those concessions, one seems to give up a lot of ground to the argument that God uses families to propagate the covenant which Abraham. If we agree that here Peter is speaking to the Jews about the covenant of Abraham, and in that these men (and we can presume that some women, I guess, without harming the text) are the fulfillment of the promises to Abraham, what's stopping us from saying, "well, dagnabit! The covenant is for parents and children together – down to a thousand generations even!"
The problem, of course, is that the covenant with Abraham in not actually the New Covenant, is it? Christ fulfills the covenant with Abraham, but He establishes a New Covenant in His own blood. So even if the Abrahamic covenant is, without any fine-tuning, a family covenant to Abraham and his fleshly kin, that's not the New Covenant which Christ established at the cross.
The burning question, then, is the one which is also asked by the advocate of this view: do we have a reasonable justification – or an express warrant, if you want to be a hard-nose – to say that the way God hands down the covenant has changed from the old to the new?
It is a great question. I have an answer on my other laptop (the good one, the Mac), and I will get it up here as soon as I smooth it out.