God said He made Ezekiel a prophet to the "house of Israel"...
Sure. Absolutely.
...there will be joined against God's people, of both ISRAELS, ...
There is only one Israel of God:
"Lest you be wise in your own sight, I do not want you to be unaware of this mystery, brothers: a partial hardening has come upon Israel, until the fullness of the Gentiles has come in. And in this way all Israel will be saved..." (Romans 11:25-26)
"Long ago, at many times and in many ways, God spoke to our fathers by the prophets, but in these last days He has spoken to us by His Son, whom He appointed the heir of all things, through whom also He created the world." (Hebrews 11:1)
This is primarily the problem of premillennialism, and particularly dispensational premillennialism. Regarding Ezekiel 37 and the valley of dry bones, God Himself has just said through Ezekiel ~ in Ezekiel 36:
"Therefore say to the house of Israel..." (that's THE house of Israel; there is only one, and not two) "...I will take you from the nations and gather you from all the countries and bring you into your own land. I will sprinkle clean water on you, and you shall be clean from all your uncleannesses, and from all your idols I will cleanse you. And I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put My Spirit within you, and cause you to walk in My statutes and be careful to obey My rules."
And Ezekiel 37 is a picture of what Paul talks about in Ephesians 2:1-10...
"And you were dead in the trespasses and sins in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience ~ among whom we all once lived in the passions of our flesh, carrying out the desires of the flesh and the mind, and were by nature children of wrath, like the rest of mankind. But God, being rich in mercy, because of the great love with which He loved us, even when we were dead in our trespasses, made us alive together with Christ ~ by grace you have been saved ~ and raised us up with Him and seated us with Him in the heavenly places in Christ Jesus..."
Paul puts the unification of God's one people, His Israel, this way, speaking to Gentiles:
"...remember that at one time you Gentiles in the flesh, called 'the uncircumcision by what is called the circumcision, which is made in the flesh by hands ~ remember that you were at that time separated from Christ, alienated from the commonwealth of Israel and strangers to the covenants of promise, having no hope and without God in the world. But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. For He Himself is our peace, who has made us both one and has broken down in His flesh the dividing wall of hostility by abolishing the law of commandments expressed in ordinances, that He might create in Himself one new man in place of the two, so making peace, and might reconcile us both to God in one body through the cross, thereby killing the hostility. And He came and preached peace to you who were far off and peace to those who were near. For through Him we both have access in one Spirit to the Father. So then you are no longer strangers and aliens, but you are fellow citizens with the saints and members of the household of God, built on the foundation of the apostles and prophets, Christ Jesus Himself being the cornerstone, in Whom the whole structure, being joined together, grows into a holy temple in the Lord. In Him you also are being built together into a dwelling place for God by the Spirit." (Ephesians 2:11-22)
And who are the true Jews of God? Well Paul tells us in Romans 2:28-29:
"For no one is a Jew who is merely one outwardly, nor is circumcision outward and physical. But a Jew is one inwardly, and circumcision is a matter of the heart, by the Spirit, not by the letter. His praise is not from man but from God."
There is only one true Israel, and it consists now of a great many people past and present of every tongue, tribe, and people group. The millennium is now, and what Ezekiel gives us a picture of in Ezekiel 37 ~ bringing dead people from death to life ~ is a present reality... it is happening right now, all around us. God is doing this.
Gloria in excelsis Deo.
Grace and peace to all.