This question is a bit of a problem for Catholics, because their councils come along over a thousand years after the writing of scripture and require you to believe some things not explicitly taught in scripture and anathematize dissenters. But does scripture explicitly teach what is necessary for salvation? Arminius said yes. Let’s look at his reasons why.
Broadly, his argument is in three steps 1) God perfectly inspired all things we need to know for salvation to the prophets and apostles, 2) they faithfully communicated them, and 3) they wrote them down in scripture. (link)
God perfectly inspired all things we need to know for salvation to the prophets and apostles
Arminius argues that John 14:26, John 15:15, John 16:13, and John 17:8 teach Christ knew and revealed all things necessary for salvation to His Apostles and he fortifies his argument based on John 17:17-20 that this revelation was intended for the sanctification of the whole church. Based on 1 Corinthians 2:6-16, Arminius argues that since gospel was predestined (v 7) it is therefore eternal and immutable (as opposed to changing over time).
Based on John 7:38-39, John 16:7-8, 13-15 and Acts 2:16-17, 28, 33, Arminius argues that the Holy Spirit will faithfully reveal the things of Christ, rather than new revelations necessary for salvation after the time of the apostles. Based on 2 Cor 3:6, Gal 3:15, Heb 8:13, Acts 2:17, John 4:25, Heb 1:2, 1 Peter 1:20 Mt 24:14, Mt 28:20, Arminius argued that since the Apostles were ministers of a new covenant revealed during the last days (which could not change or be added to) the revelation given to them contained all things necessary for salvation.
They faithfully communicated them
Based on John 1:18, 3:32, 1 John 1:3, 1 Cor. 2:6-16 Arminius argues that Christ and the Apostles declared what they had seen and heard. Based on Romans 10:8-10, 1 Cor 1:21, 2 Cor 3:9, Acts 13:26, Romans 1:16, and 1 Peter 1:23-25, Arminius argues that the Gospel contains the whole message needed for salvation, and based on 2 Cor 3:6, Col 1:24-28 he argues that the Apostles were faithful ministers of the Gospel.
Based on Luke 7:30, Acts 20:27 and 1 Cor 15:1-2, Arminius argues that all the counsel of God contains the teachings necessary for salvation and that all the council of God was declared by the Apostles. Then Arminius answers the objection that the Apostles taught what was necessary at the time, but not what was necessary to the end of the world, by citing Gal 1:7-9 as prohibiting adding to the Gospel and based on Eph 2:20-21 which states the whole church (not just part of it) was built on the foundation of the Apostles.
They wrote them down in scripture
Based on Deut 4:2,12:28, 30:10-14, 28:58, Josh 1:7-8, 1 Cor 4:6, Acts 20:27, 26:22, Arminius argues the scriptures claim their completeness and prohibit additions.
Based on Tit 1:1, 1 Chron 28:9, John 5:23, 17:3, Arminius argues that saving doctrine consists of knowing God and Christ. Based on 1 John 5:13; Timothy 3:16; Rom. 15:4; 1 Thess. 1:3 and Tit. 2:12-13, Arminius argues that scripture perfectly delivers that doctrine. Based on 2 Cor. 3:6; Gal. 3:15; Jer. 31:31-34; 32:38-40; Gal. 4:1-2, Arminius argues that the Old and New Testament are a Testament or Covenant and therefore complete and not to be added to. Based on Luke 16:16; Josh 1:8; Luke 1:1-4; Rom. 1:2-6; Acts 26:22-23, Arminius argues that the Law and Gospel are contained in scripture and contain the whole of what we need to know for salvation.
Based on Romans 10:4-10, 1 John 3:23, John 20:31, 1 John 5:9-13, Mt 22:37-40, John 5:39, Luke 16:27-30, and 2 Tim 3:15-17, Arminius argues that scripture states its goal as providing saving doctrine. This is confirmed because the scriptures use the terms “prophets” interchangeably with “prophetic scriptures” and likewise with God and scripture. (Acts 26:27, Luke 16:29, 2 Peter 1:19-20, Luke 24:27, Romans 9:17, Exod 9:16, Gal 3:22, Romans 11:32, Gal 3:8, with Gen 12:2-3)Based on John 5:39; Rom. 1:3; Luke 24:27, Arminius argues that the scriptures contain all things necessary to be known for the salvation of the Church.
Arminius concludes that based on 2 Tim. 3:16; Matt. 4:3-4; 22:29 Acts 18:28, the scriptures teach all things that have been or will be necessary for salvation. Thus, all errors regarding salvation are refutable by scripture. If something isn’t in scripture, it’s not necessary for salvation. If something can’t be refuted by scripture, it’s not heretical. When Catholics, without scriptural proof, either add doctrines as necessary or anathematize doctrines scriptures do not anathematize, we can’t join them.