Some believe that the church has replaced Israel or is the “New Israel,” so is the church the New Israel?
Israelite Feast Days
I know of several churches that claim we are still supposed to keep the feast days of the Old Testament as well as observe the Sabbath as Saturday, however there is no such teaching in the New Testament, and besides, the church apparently gathered on the first day of the week, which is Sunday. Even so, many churches believe in tithing (some up to 30%), observing the Jewish feast days, and some even avoid what they deem as “unclean” foods, but my question is this: Are they Jewish, because these laws were written to the nation of Israel. Even in the Sabbath command, it was to be a sign between God and Israel, and always included words such as, “And to you O Israel” (Ex 31:13, 16, 35:1), or something to that effect, so these days or laws were not meant for Gentiles to observe. They were a foreshadowing of the coming Messiah, Jesus Christ, Who would fulfill the meaning of these days. Even if you are Jewish, with the redemptive work of Christ already completed, there is no need to observe these things. We are to gather together with the Body of Christ to do the things commanded by Christ (Matt 25:35-36, 28:18-20), “Therefore let no one pass judgment on you in questions of food and drink, or with regard to a festival or a new moon or a Sabbath. These are a shadow of the things to come, but the substance belongs to Christ” (Col 2:16-17). The church is not Israel. God has a purpose and plan for Israel, but also for those who’ve trusted in Christ. God’s will for the Jews is the same for everyone; believe on the Lord Jesus Christ and be saved (Acts 4:12, 16:30-31).
Chosen by God
The nation of Israel was called by God out of their Egyptian bondage, but God had long ago promised Abraham that He would make of him a great nation, and they did end up as a great nation; at least as far as there size was concerned. Pharaoh was so concerned about the population of the Israelites that he ordered all the Israelite babies killed at birth, but it was God Who brought Israel out of Egypt with a mighty hand, performing miraculous wonders. Israel had nothing to offer God and they could not escape Egypt on their own. It was only for the sake of covenant that He made with Abraham that God brought Israel out of slavery, and in time, into the Promised Land. They couldn’t have delivered themselves, just as believers couldn’t quicken themselves to a new life in Christ without the Spirit of God (Eph 2:1-5). And like ancient Israel, God did not call us because we were good or lovely; God called us because He first loved us (Eph 2:4-5; 1 John 4:19). God told Israel, “It was not because you were more in number than any other people that the LORD set his love on you and chose you, for you were the fewest of all peoples” (Deut 7:7), so it was “because the LORD loves you and is keeping the oath that he swore to your fathers, that the LORD has brought you out with a mighty hand and redeemed you from the house of slavery, from the hand of Pharaoh king of Egypt” (Deut 7:8).
The idea that the church has replaced Israel is called Replacement Theology or supersessionism. It holds that the Christian church has succeeded where the nation Israel had failed, and Israel used to be called the people of God, but now, the people of God are those in the Body of Christ, the church. Some even believe that the people of Israel are no longer God’s children, and now that Israel is no longer doing the work of God, the church has taken up the mantel and taken over where Israel failed. But what about the promise made to Abraham? Is God really finished with Israel? Absolutely not! Anyone that’s witnessed to a Jewish person realizes that they do not accept Jesus as the Messiah, or at least not yet. In the context of Israel’s rejection of Christ, the Apostle Paul says that “a partial hardening has come upon Israel, until the fullness of the Gentiles has come in” (Rom 11:25). Paul is referring to the time when “The Deliverer will come from Zion [and] he will banish ungodliness from Jacob; and this will be my covenant with them when I take away their sins” (Rom 11:26-27). Notice that it’s a “partial hardening” and not a permanent one, and Jacob never refers to believers but always refers to the nation of Israel. God changed Jacob’s name to Israel, so when you read something in the Bible about Jacob, it is also referring to the nation of Israel. At that time (Rom 11:26-27), many of the Jews will finally believe that Jesus is not just a Prophet, but is in fact, the Messiah, and God of very God. He will bring with Him a new covenant that takes away their sins (the one Christians now are under), so it sure doesn’t sound like God is finished with the Jews. The time is coming when many will finally believe; it’s just not today.
There are a few Messianic Jews who have already put their trust in the Savior, but most Jews still reject Jesus as the Messiah, or as God, but just because the Jews don’t believe in Jesus right now does not mean they won’t believe in Him someday. God has a different time table for the Jews than for the church, but today is the day of salvation for Jew and Gentile (2 Cor 6:2). Everyone needs to repent and believe the gospel, just as Jesus said (Mark 1:14-15).
Here is some related reading for you: Why Should We Pray for Israel? A Christian Perspective 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.