When a Christian sins, are they automatically rejected by God? Does this mean they are no longer a Christian until they repent?
Men put up a barrier between themselves and God with their sins. "Behold, the LORD'S hand is not shortened, that it cannot save; nor His ear heavy, that it cannot hear. But your iniquities have separated you from your God; and your sins have hidden His face from you, so that He will not hear" (Isaiah 59:1-2). The problem is men rejecting God, not God rejecting men.
When an Israelite sinned, he did not stop being an Israelite. He was under a covenant with God that lasted his entire lifetime. He might not be accepted by God because he won't let go of his sins, but if he does, it did not mean he had to become an Israelite all over again.
When someone becomes a Christian, he too has entered into a lifelong covenant with God. "In Him you were also circumcised with the circumcision made without hands, by putting off the body of the sins of the flesh, by the circumcision of Christ, buried with Him in baptism, in which you also were raised with Him through faith in the working of God, who raised Him from the dead. And you, being dead in your trespasses and the uncircumcision of your flesh, He has made alive together with Him, having forgiven you all trespasses" (Colossians 2:11-13). In doing so, he becomes a child of God. "For you are all sons of God through faith in Christ Jesus. For as many of you as were baptized into Christ have put on Christ" (Galatians 3:26-27).
A Christian can sin, straying from the truth, but it doesn't mean he ceases to be a Christian. "And if anyone does not obey our word in this epistle, note that person and do not keep company with him, that he may be ashamed. Yet do not count him as an enemy, but admonish him as a brother" (II Thessalonians 3:14-15). If a sinning Christian returns to the truth, he doesn't have to become a Christian all over again because he never stopped being a Christian. "If we say that we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, He is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say that we have not sinned, we make Him a liar, and His word is not in us. My little children, these things I write to you, so that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous" (I John 1:8-2:1).
Of course, the world is full of people who claim to be Christians but who have never done as the Lord commanded. Since they never truly left their sins in the first place, they would need to become a child of God in truth to be freed from their sins (Romans 6:3-7).
Thanks for your time and sharing the Word of God with me. May God continue to bless you.