There is no movement called "the Way", I only gave it to you because you asked for a name. The early Jewish disciples used that name because they were zealous for the Torah and the "way" is a reference to faithful Torah obedience.
Let me give you a bit of an overview.
Covenants are promises. They are made between two parties and are valid as long as both of them are alive / exist. Newer covenants may build on an existing one, but it never replaces it. Therefore, the new covenant of Jesus doesn't replace the covenant God made with Israel, which is, "You are my people and I am your God". The Torah describes how you live within that covenant, that is, it describes the covenant relationship.
A covenant cannot be broken, only the covenant relationship can be, which needs to be renewed. This way, when the sinful of Israel return to God they renew their covenant relationship with God.
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When the sinful of Israel return to God ?? The so called sinful are not Israel and never were. as it's the dogs that are out side is it not. they are just Jews.
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Jeremiah's new covenant is not a brand new covenant with a brand new set of laws, but a placing of the existing laws within the hearts of the Jewish people. In that sense, it is still the existing covenant with a new way of giving of the Torah (ie, not in written form that people had to internalise - put in the heart).
The covenant of the Jews go back to Abraham's promise about the seed in whom all families of the world will be blessed. Gentiles were never under this covenant, but under the one God made with Noah. Hence, Melchizedek was the high priest of God within that covenant, in which the laws of Noah applied.
Now Jesus utilises another covenant, which seems to go back to an earlier promise God made to Abraham, that he shall be the father of many nations. This includes both Jews and Gentiles who mimic his faithfulness. Jews were already in that covenant because the other Abrahamic and the Mosaic covenants build on it (their covenant even includes the Noahide and Adamite covenants).
But Gentiles before were not God's people and are now called into this covenant by a man who showed what faithful life they were expected to live. Those who immerse themselves into the faithful life of Jesus to the point of death, that is, they repent (die to the old sinful life) and mimic live the righteous life Jesus lived, are the ones who died to their old man and being born again live in the newness of life. The entry into this covenant is by repentance and faith, which faith (the faith of Abraham) which includes righteous conduct. This is how Gentile converts are drawn near and become partakers in this covenant.
There are two views regarding what laws are required of Gentile converts:
1. the Jerusalem council lists the laws of Noah with additional laws may be required depending on local conditions (the reason given is that Moses has been preached in their cities).
Acts 15:19 Therefore it is my judgment that we do not trouble those who are turning to God from among the Gentiles, 20 but that we write to them that they abstain from things contaminated by idols and from fornication and from what is strangled and from blood. 21 For Moses from ancient generations has in every city those who preach him, since he is read in the synagogues every Sabbath.”
Acts 21:25 But concerning the Gentiles who have believed, we wrote, having decided that they should abstain from meat sacrificed to idols and from blood and from what is strangled and from fornication.”
2. the Jerusalem council lists the laws of the "ger" from Lev 17-18 as core laws required for peaceful co-existence between Jewish and Gentile believers. Gentile believers then would receive the full Torah training over the years and could voluntarily convert.
If you ask my opinion, these are basic laws that were expected from all Gentiles to be counted righteous, independently of their beliefs in Jesus, whether these are derived from Lev 17-18 or from the Noahide laws, but Gentile converts were encouraged to observe as much of the Torah as they could. Hence Paul could write:
Romans 2
14 For when Gentiles who do not have the Law do instinctively the things of the Law, these, not having the Law, are a law to themselves, 15 in that they show the work of the Law written in their hearts, their conscience bearing witness and their thoughts alternately accusing or else defending them
...
25 For indeed circumcision is of value if you practice the Law; but if you are a transgressor of the Law, your circumcision has become uncircumcision. 26 So if the uncircumcised man keeps the requirements of the Law, will not his uncircumcision be regarded as circumcision? 27 And he who is physically uncircumcised, if he keeps the Law, will he not judge you who though having the letter of the Law and circumcision are a transgressor of the Law? 28 For he is not a Jew who is one outwardly, nor is circumcision that which is outward in the flesh. 29 But he is a Jew who is one inwardly; and circumcision is that which is of the heart, by the Spirit, not by the letter; and his praise is not from men, but from God.
Paul is not saying that Gentiles become Jews by observing the Torah, he only illustrates that it is not circumcision, but obedience that matters. Indeed, physical circumcision carries the symbolic meaning of repentance - putting away the old flesh, cutting it off, and you cannot take it back.
The Noahide laws fall into 7 categories:
Prohibition of Idolatry
Prohibition of Murder
Prohibition of Theft
Prohibition of Sexual immorality
Prohibition of Blasphemy
Prohibition of eating flesh taken from an animal while it is still alive (you can knock it unconscious, the only way you know it is dead is if you pour out all of its blood)
Establishment of courts of law
You notice that the letter of the council doesn't mention theft or murder. Does that mean these were OK? I doubt. My take is that the council's letter simply mentioned the core laws that Gentiles were generally guilty of breaking, therefore they needed to be emphasised. They were certainly expected to observe the full list of the laws of Noah.
Gen 9:5
...And
from every man,
from every man’s brother I will require the life of man.
This is the prohibition of murder and the prohibition of assisted suicide (euthanasia).
So the conclusion is that no, Gentiles didn't have to convert to Judaism in order to be considered saved and be counted as the people of God. But since the Torah was given as the light to the world, and those "who lived in great darkness saw a great light", we are encouraged to observe as much of it as we can (without coming under its condemnation if we don't - you remember, we are not obliged, but encouraged because these are the laws of God) and which helps to purify our walk in this world. "By the Torah is the knowledge of sin", therefore, we should reflect on it to learn the ways of God.
Not all laws are applicable. If you are a free male, you don't observe the laws of women, slaves, priesthood and kings. So you only ever have a subset of laws that apply. You should notice that even though Paul is very much against getting the Gentiles circumcised for salvation, he applies the laws of the Torah way beyond what is required by the laws listed by the council.
To sum it up, Jews continued to have the obligation to observe the Torah completely. They were already under these covenants, so they only renewed their covenant relationship with God. Gentile sinners had no such promises, though they were under the Noahide covenant (as all Gentiles are), they have forgotten God and their covenant. When they convert they enter into the earlier Abrahamic covenant and are obliged to observe the required laws, as well as encouraged to observe the Torah, the "good word of God".
One is saved from his sin when he repents and immerses himself in the faithful life of the Messiah, ie, lives faithfully (walks as he also walked"). Belief alone is worthless unless are coupled with all these. God always and ever cared about conduct, not head knowledge.
So much for now.