Is repentance necessary for salvation?
It depends on what you mean by repentance.
If you mean 1) that you must stop sinning before coming to Christ then obviously no. But if you mean 2) knowing you cannot save yourself, by grace, you turn from your sin to trust in Christ to deliver you from it, then yes, it is necessary. No one can come to Christ unless they come to Him as the Savior from sin.
Evangelical repentance is ... a gracious principle and habit implanted in the soul by the Spirit of Christ, in the exercise of which a regenerate and believing sinner, deeply sensible of the exceeding sinfulness and just demerit of his innumerable sins is truly humbled and grieved before the Lord, on account of the sinfulness and hurtfulness of them. He feels bitter remorse, unfeigned sorrow, and deep self-abhorrence for the aggravated transgressions of his life, and the deep depravity of his nature; chiefly, because by all his innumerable provocations he has dishonoured an infinitely holy and gracious God, transgressed a law which is "holy, and just, and good," and defiled, deformed, and even destroyed his own precious soul. This godly sorrow for sin and this holy abhorrence of it arise from a spiritual discovery of pardoning mercy with God in Christ, and from the exercise of trusting in His mercy. And these feelings and exercises are always accompanied by an unfeigned love of universal holiness, and by fixed resolutions and endeavours to turn from all iniquity to God and to walk before him in newness of life. Such, in general is the nature of that evangelical repentance, to the habit and exercise of which the Lord Jesus calls sinners who hear the Gospel."
- John Colquhoun
1. Repentance is taking full responsibility for the sin
2. Repentance is turning from the idol we serve to the true God
What do we mean by idols? How does that tie into sin? Sin is not just a transgression of a law of God. It involves a turning from God in unbelief and turning to something else as a "god". When I sin I am saying to God, "I do not trust you. I do not believe your way is good and best. I do not believe you are wise." In place of the true God I worship pleasure, a lover, a lie, my money, a career advancement, my reputation etc. We cannot worship God and sin. we cannot sin without worshiping idols. Deep in the heart of man there is a powerful pull of idolatry. We want to worship this other god because of the pleasure it brings us. That love for sin and out false lover can actually keep us from repentance.
- Mark Lauterbach - The Transforming Community 142-144