While Paul uses a lot of ink arguing for faith rather than works, he never really argues that faith in God without faith in Jesus is insufficient. To modern ears, that seems to be the obvious question.
We know that Paul believes this because his entire life as an apostle was dedicated to teaching Jews and God-fearing Gentiles to believe in Jesus. Why bother if faith in God would have been enough? Why bring salvation to people who already believed in God if they were already saved?
So what is it about Jesus that makes faith in him essential? Well, consider what Paul says in 1 Cor, perhaps his earliest surviving letter.
(1Co 1:2 ESV) To the church of God that is in Corinth, to those sanctified in Christ Jesus, called to be saints together with all those who in every place call upon the name of our Lord Jesus Christ, both their Lord and ours:
Those who “call upon the name of our Lord Jesus Christ (Lord Jesus, the Messiah)” is another obvious reference to Joel 2:32, which promises salvation to all who call upon the name the “Lord,” that is, YHWH. Paul is also calling Jesus YHWH!
He makes the clear all the more clear in chapter 8 —
(1 Cor 8:5–6 ESV) 5 For although there may be so-called gods in heaven or on earth—as indeed there are many “gods” and many “lords”— 6 yet for us there is one God, the Father, from whom are all things and for whom we exist, and one Lord, Jesus Christ, through whom are all things and through whom we exist.
Paul insists on faith in the one-God of Judaism — but only as understood through Jesus.
Paul’s point is not just that there is a single deity. Staked out here is one’s relation to that deity, the characteristics of that deity, and what we know about that deity–these are what really matter. Life must be lived in light of the deity who is in control (cf. 3:21–23). It is not an option to be unrelated to that God. Life must be lived, moral decisions must be rendered in keeping with what we know about that God. The nature of all being is imaginable only within the scope allowed by and dictated by the nature and purpose of God.
A fundamental issue is engaged here: How does what one believes and knows about God bear on the relatively mundane decisions of everyday life? Paul’s answer: absolutely critically and definitionally. To whom does one belong? What are the characteristics of the one to whom one belongs? These are recurring issues in 1 Corinthians because they are fundamental to living the life of faith, to understanding one’s place in the world, and to moral reasoning.
J. Paul Sampley, “The First Letter to the Corinthians,” in The Acts of the Apostles-The First Letter to the Corinthians (vol. 10 of New Interpreters Bible, Accordance electronic ed. Nashville: Abingdon Press, 2002), n.p.
We know that there is in fact only one true God and one true Lord. The key words of v. 6, “Lord,” “God” and “one,” are taken from Deuteronomy 6:4 (“the Lord our God, the Lord is one”), in which Lord and God both refer to the same (one) God. Here Paul “has glossed ‘God’ with ‘the Father’, and ‘Lord’ with ‘Jesus Christ’, adding in each case an explanatory phrase: ‘God’ is the Father, ‘from whom are all things and we to him’, and the ‘Lord’ is Jesus the Messiah, ‘through whom are all things and we through him.’” Paul thus simultaneously reaffirms strict Jewish monotheism and the highest possible Christology imaginable. Christ finds his identity within the very definition of that one God/Lord of Israel.
Roy E. Ciampa and Brian S. Rosner, The First Letter to the Corinthians (Pillar NTC; Accordance electronic ed. Grand Rapids: Eerdmans, 2010), n.p.
Amazingly, in 1 Cor 8:6-8, Paul takes the Jewish Shema (“Hear, O Israel, the Lord your God is One God …”) and turns it into a declaration that Christians worship but one God, but that the One God is both God the Father and the Lord Jesus Christ.
In Paul’s mind, God has now revealed himself as both God the Father and Jesus who both Lord and Messiah. Believing this is “faith” and nothing else will do. Again: you can’t enter a Kingdom if you deny the King.