Not true. Jeremiah 31:31-32 says that the new covenant is "not like the covenant which I made with their fathers in the day I took them ... out of the land of Egypt." - which is the Mosaic Law. So no, the new covenant put in our hearts is not the Mosaic Law.
For those that don't know...
In the Old Testament the Law is type and shadow of the Holy Spirit. It is HE Whom God promised Israel to put in their inward parts to lead, guide, convict, speak to except instead of doing it in tablets of stone and from without God does it on fleshly tables of the 'heart' and from within.
And yes, the new covenant does apply to Gentile believers.
Show me in Jeremaih 31:31-34 where Gentiles are identified. Because from what I read the New Covenant is between God and the House of Israel (northern kingdom) and Judah (southern kingdom.) There are no Gentiles even remotely named or identified. You have too much Gentile interpretation in you. Read the Hebrew Scriptures like a Hebrew - if that's even possible with your bias - and come to the knowledge of the truth.
For He Himself is our peace In Christ, "there is neither Jew nor Greek, there is neither slave nor free man, there is neither male nor female; for you are all one in Christ Jesus. And if you belong to Christ, then your are Abraham's descendants, heirs according to the promise." Galatians 3:28-29
Why do so many Gentiles miss the truth of Saul's words.
The Galatian letter as with all the New Covenant writings are written to Jewish Christians. Chapter three Saul is addressing Jewish Christians and addressing the Law, their history, prophecies, their religion, and the effect of Israel's Messiah on the prior covenants (Abrahamic and Mosaic) and their concern as to their standing and relation to those covenants now that Messiah has come and they have had experience with the Holy Spirit of Promise. Saul says it clearly:
23 But before faith came,
we [Jews] were kept under the law, shut up unto the faith which should afterwards be revealed.
24 Wherefore t
he law was our [Jews] schoolmaster to bring
us [Jews] unto Christ, that
we [Jews] might be justified by faith.
25 But after that faith is come,
we [Jews] are no longer under a schoolmaster.
26 For
ye are all [Jews] the children of God by faith in Christ Jesus.
27 For as many of
you [Jews] as have been baptized into Christ have put on Christ.
28 There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for
ye [Jews] are all one in Christ Jesus.
29 And if
ye [Jews] be Christ’s, then are
ye [still Jews and] Abraham’s seed, and
heirs according to the promise.
Gal. 3:22–29.
The promise is to Abraham and to his seed and Gentiles are not his seed. They have never been "heirs" with Abraham or his seed (Israel). Never. Not then, and definitely not now. It's a Jewish Covenant with Jewish history, culture, religion, persons, prophecies, promises, and Jewish Messiah. So, let's leave them to what belongs to them and not try to steal THEIR inheritance according to the promise God gave to Abraham.
So Gentile believers are not only heirs of the Abrahamic Covenant but also of the New Covenant.
There is nothing of the sort. Gentiles are not even named or identified in the Abrahamic (Gen. 17) and New Covenants (Jer 31.) You've been taught too much Gentile error and lies. Read the Scripture and come to the knowledge of the truth.
"I have other sheep, which are not of this fold (Gentiles); I must bring them also, and they will hear My voice; and they will become one flock with one shepherd." John 10:16
The other sheep are Samaritans. NOT Gentiles.
The Jews are no longer God's chosen people. Those in Christ, Jew and Gentile, "are a chosen race, a royal priesthood, a holy nation, a people for God's own possession." 1 Peter 2:9
Peter is quoting Deuteronomy to Jewish Christians. They alone would understand the significance of what Peter is addressing in his two epistles.
Jesus loved the Gentiles as well as the Jews. He wanted to bring them into the fold. So he reached out to the Samaritan woman in Sychar and she was used by God to bring many from that city to their Messiah. The Samaritan's, who were part Jew, part Gentile, came to understand that Jesus was "indeed the Christ, the Savior of the world", not just the Jews. John 4:42
The Samaritan was half-Jew at least but still in Covenant with God.
If the woman was truly Gentile she would have worshiped their millions of gods and idols. Read the passages. She was a daughter of Abraham and waited for Messiah.