Isn't the Church Israel?
Simply put, no. Although there is a traditional stance among the established churches that teaches otherwise (commonly known as 'replacement theology' - where ever one reads Israel it means the Church) there is actually nowhere in the New Testament that supports this misguided teaching. There are however many passages that clearly refute replacement theology:
Brothers, my heart's desire and my prayer to God is for Israel, that they may be saved.
Rom. 10:1
1 I ask then, did God reject his people? May it never be! For I also am an Israelite, a descendant of Abraham, of the tribe of Benjamin.
25 For I don't desire you to be ignorant, brothers, of this mystery, so that you won't be wise in your own conceits, that a partial hardening has happened to Israel, until the fullness of the Gentiles has come in, 26 and so all Israel will be saved. Even as it is written,
"There will come out of Zion the Deliverer,
and he will turn away ungodliness from Jacob.
27 This is my covenant to them,
when I will take away their sins."
Rom. 11:1,25-27
From just these verses it is obvious that there is a difference between Israel and the Church (Paul is addressing believers and therefore the Church) - chapter nine of Romans makes the same distinction. Consider also what God says through the prophet Jeremiah:
35 Thus says the LORD, who gives the sun for a light by day, and the ordinances of the moon and of the stars for a light by night, who stirs up the sea, so that its waves roar; the LORD of hosts is his name: 36 If these ordinances depart from before me, says the LORD, then the seed of Israel also shall cease from being a nation before me forever. 37 Thus says the LORD: If heaven above can be measured, and the foundations of the earth searched out beneath, then will I also cast off all the seed of Israel for all that they have done, says the LORD.
Jer. 31:35-37
So then, that whilst it is true that the Church does receive the blessings and promises of God, it is not true that the Church has supplanted Israel as the 'people of God' and therefore entitled to 'inherit' the name.
Contrary to popular thinking, a Jew who accepts Jesus as Messiah does not join the [Gentile] Church,1; in actual fact a Gentile who accepts Christ joins together with the Jewish believers:
11 Therefore remember that once you, the Gentiles in the flesh, who are called "uncircumcision" by that which is called "circumcision," (in the flesh, made by hands); 12 that you were at that time separate from Christ, alienated from the commonwealth of Israel, and strangers from the covenants of the promise, having no hope and without God in the world. 13 But now in Christ Jesus you who once were far off are made near in the blood of Christ.
19 So then you are no longer strangers and foreigners, but you are fellow citizens with the saints, and of the household of God,
Eph. 2:11-13,19
16 If the first fruit is holy, so is the lump. If the root is holy, so are the branches. 17 But if some of the branches were broken off, and you, being a wild olive, were grafted in among them, and became partaker with them of the root and of the richness of the olive tree; 18 don't boast over the branches. But if you boast, it is not you who support the root, but the root supports you.
24 For if you were cut out of that which is by nature a wild olive tree, and were grafted contrary to nature into a good olive tree, how much more will these, which are the natural branches, be grafted into their own olive tree?
Rom. 11:16-18,24
Notice who the olive tree represents (v. 24) and that Christians are grafted into it. This then makes [Gentile] Christians members of Israel - notice that they are members of, and not Israel itself.
1 The Church is not a Gentile invention as it were (up until the establishment of the Church at Antioch, Acts 11:20-26, the vast majority of the members of the Church were Jewish), but rather a term [mostly] used to describe God's people. The Greek word used is 'ekklesia' pronounced ek-klay-see'-ah and according to Thayers is:
1) a gathering of citizens called out from their homes into some public place, an assembly
1a) an assembly of the people convened at the public place of the council for the purpose of deliberating
1b) the assembly of the Israelites
1c) any gathering or throng of men assembled by chance, tumultuously
1d) in a Christian sense
1d1) an assembly of Christians gathered for worship in a religious meeting
1d2) a company of Christians, or of those who, hoping for eternal salvation through Jesus Christ, observe their own religious rites, hold their own religious meetings, and manage their own affairs, according to regulations prescribed for the body for order's sake
1d3) those who anywhere, in a city, village, constitute such a company and are united into one body
1d4) the whole body of Christians scattered throughout the earth
1d5) the assembly of faithful Christians already dead and received into heaven
The Church then consists of all those that have accepted Jesus as their Lord and Savior including both Jews and Gentiles.
Whilst it is certainly true that we as Christians are now receiving the spiritual blessings first promised to Israel, Israel will, after recognizing and accepting the Messiah, receive all that which was promised. The Church has in no way usurped Israel and certainly has not inherited the name. God still has a plan for Israel and even now is bringing it to pass.2
2 Whose Land Is It?, Why Pray for Israel? and also 1967 - A Fulfillment of Prophecy?