No, where does He say that one can go on sinning and be saved? Luther's whole idea that one is "saved" by the sacrifice on the cross because it outweighs all the sins of the world is where his fetal error (no pun intended) is. Christ made it possible for all to be saved, but that certainly is not accomplished by sinning boldly or by "thousands of fornications a day!"
Contrary to Paul, and Luther, Christ teaches works-based salvation, works based on faith. Our conduct does count. He forgave sins and said "sin no more." Our intent is to not want to sin any more. But satanic Luther's theology teaches that one can go on sinning and still be saved, because Luther says that Paul says so!? What does Christ say about that?
And as for sin separating us from God, that's a given. Death came into the world through sin, because sin is separation from God who is life. The OT reminds us that the soul "that sinneth dies." Christ came into the world and freed us from death "because all have sinned." He gave us a parole, not a clean bill of health.
He says (John 13:10), “He that is washed needeth not save to wash his feet, but is clean every whit: and ye are clean, but not all”
First, you have changed the question. Your original statement was “If sin doesn’t separate us from God, what does?” My answer was, “For the believer, I guess you would have to ask Paul” Your statement was, “Paul? Why, I would rather ask Christ. And what Paul says, Christ never said” and I said, “Where in the gospels does Christ say anything about sin separating one from God or the Father?”
Jesus does not address sin separating us from the Father, Paul does. What Paul is addressing in his letter to Melenchton is similar to the problem John is addressing in his first epistle. (1 John 1:8-10) “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.
“If you are a preacher of mercy, do not preach an imaginary but the true mercy. If the mercy is true, you must therefore bear the true, not an imaginary sin. God does not save those who are only imaginary sinners. Be a sinner, and let your sins be strong, but let your trust in Christ be stronger, and rejoice in Christ who is the victor over sin, death, and the world. We will commit sins while we are here, for this life is not a place where justice resides. We, however, says Peter (2. Peter 3:13) are looking forward to a new heaven and a new earth where justice will reign. It suffices that through God’s glory we have recognized the Lamb who takes away the sin of the world. No sin can separate us from Him, even if we were to kill or commit adultery thousands of times each day. Do you think such an exalted Lamb paid merely a small price with a meager sacrifice for our sins? Pray hard for you are quite a sinner.”
Luther is using hyperbole to bring home the truth that one has to admit he is a sinner, a real sinner or there is no forgiveness. He is saying there is no sin that cannot be forgiven because of the worth of the sacrifice and he is saying even after salvation we sin and the sacrifice avails for that. We all sin, but believers are reconciled to god, the relationship has been restored because of Calvary but because of our human nature we still sin and there is forgiveness. We aren't saved again every time we confess.