Will God save Israelites simply because they are his chosen people even if they are not righteous?

In the New Testament, to be saved means to be forgiven your sins and given the hope of eternal life. This is only possible through belief in Jesus Christ and baptism into his name: John 3:16, Mark 16:16, Acts 4:12. Israelites have the same opportunity to obtain salvation in this way as the people of all the other countries of the world.

There is neither Jew nor Greek, there is neither slave nor free, there is no male and female, for you are all one in Christ Jesus. (Galatians 3:29)

(Also see, ‘Do Israelites who do not and will not believe in Christ go to paradise just because they are from Israel?‘.)