Both Jewish and Gentile Christians today make up the "Church", the "body of Christ" or "one new man", which Paul discusses in Ephesians 2. We, as Gentiles, were aliens from the commonwealth of Israel and strangers from the covenants of promise (Eph. 2:12), but now we have been brought near through Christ. But notice that we are not said to be "part of Israel" or a "new Israel." Instead, both Jews and Gentiles are said to be joined into a new spiritual body, the one new man (Eph. 2:15).Another evidence that we are not "part of Israel" is found by carefully examining Paul's use of the term "Israel" in Romans 9-11, and especially chapter 11. There, it will be found that "Israel" still refers to Jews.
What can be confusing is that although believing Gentiles are considered to be "sons of Abraham" by faith (Gal. 3:7, 29; Php. 3:3), they are never called "Israel." A distinction is made between being blessed as part of the Abrahamic covenant — rooted in the promises to Abraham — which included blessings for "all the families of the earth" (wherein Gentile believers are found) and blessings for the physical descendants of Abraham through Isaac and Jacob/Israel — the Jews.
In an analogous way to the way that Ishmaelites and Edomites are "sons of Abraham" physically, but not Israel, believers are spiritual sons of Abraham, but not Israel.