Is salvation possible without the Spirit?
No. The Spirit leads us through faithful response to the gospel into the fellowship of the Church.
Catechism for the Global Methodist Church, question 23
3Jesus answered and said to him, "Truly, truly, I say to you, unless one is begotten from above, that one cannot see the kingdom of God." 4Nicodemus said to him, "How can a human be begotten when he is an old man? He cannot enter his mother's womb a second time and be begotten." 5Jesus answered, "Truly, truly, I say to you, unless one is begotten out from water and spirit, that one cannot enter the kingdom of God." 6What is begotten out from the flesh is flesh, and what is begotten out from the spirit is spirit.
9You, however, are not in the flesh but in the Spirit, provided that the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him. 10If Christ is in you, the body is dead because of sin, yet the Spirit is life because of righteousness. 11If the Spirit of the one who raised Jesus from the dead dwells in you, then the one who raised Christ Jesus from the dead will also give life to your mortal bodies through his Spirit dwelling in you. 12So then, brothers, we are debtors, not to the flesh, to live according to the flesh. 13For if you live according to the flesh you are about to die, but if by the Spirit you put to death the actions of the body you will live. 14For all who are led by the Spirit of God, these are sons of God. 15For you did not receive a spirit of slavery again toward fear, but you received a spirit of adoption as son, in which we cry, "Abba, Father." 16The Spirit himself testifies together with our spirit that we are children of God. 17And if children, then also heirs, heirs of God and co-heirs of Christ, if indeed we suffer with him so that we may also be glorified with him.
17And having come, he proclaimed peace to you who were far off and peace to those who were near, 18that through him we both have access in one Spirit to the Father. 19Therefore you are no longer foreigners and sojourners, but you are fellow citizens with the saints and members of the household of God, 20having been built upon the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone, 21in whom the whole building, being fitted together, grows into a holy temple in the Lord, 22in whom you also are being built together into a dwelling place of God in the Spirit.
4But when the kindness and the benevolence of God our Savior appeared, 5not by works that we did in righteousness, but according to his own mercy he saved us, through the washing of regeneration and renewal of the Holy Spirit, 6whom he poured out on us richly through Jesus Christ our Savior, 7so that, having been justified by that one's grace, we might become heirs according to the hope of eternal life.
This question follows the order of Article III of the Confession of Faith. After the Spirit convinces of sin, the article continues: "He leads men through faithful response to the Gospel into the fellowship of the Church." The catechism turns that sentence into a plain "No" and then repeats it.
Jesus said as much to Nicodemus: "unless one is begotten from above, that one cannot see the kingdom of God" (John 3:3), and again, "unless one is begotten out from water and spirit, that one cannot enter the kingdom of God" (John 3:5). Paul draws the line just as sharply: "Anyone who does not have the Spirit of Christ does not belong to him" (Romans 8:9). Titus describes the whole sequence as God's mercy: he "saved us, through the washing of regeneration and renewal of the Holy Spirit, whom he poured out on us richly through Jesus Christ our Savior" (Titus 3:5-6), so that "having been justified by that one's grace, we might become heirs according to the hope of eternal life" (Titus 3:7).
"Faithful response to the gospel" names the human side of the same work. The Spirit does not save anyone against their will; he leads, and the sinner follows in faith. That response is itself enabled by grace. ¶102 of the Book of Doctrines and Discipline explains that preventing grace "restores a measure of liberty to receive the further graces of God," so that "preventing grace enables genuine response to the continuing work of God's grace." The one who responds receives "a spirit of adoption as son, in which we cry, 'Abba, Father,'" and "the Spirit himself testifies together with our spirit that we are children of God" (Romans 8:15-16). Wesley called this the witness of the Spirit, and ¶102 says justifying grace "ordinarily results in the direct assurance" of it.
The phrase "into the fellowship of the Church" rules out private religion. Through Christ "we both have access in one Spirit to the Father," and the saved are "fellow citizens with the saints and members of the household of God" (Ephesians 2:18-19), "being built together into a dwelling place of God in the Spirit" (Ephesians 2:22). A Methodist who has been born of the Spirit will therefore be found in the congregation, at the table, and among the people the Spirit is building, because that is where the Spirit leads.