Romans 1:16-17 & John 1:12
Salvation, σωτηρίαν, and righteousness, δικαιοσύνη, are first used when Paul begins the Gospel:
16 For I am not ashamed of the gospel of Christ, for it is the power of God to salvation, σωτηρίαν, for everyone who believes, for the Jew first and also for the Greek. 17 For in it the righteousness, δικαιοσύνη, of God is revealed from faith for faith, as it is written, “The just shall live by faith.” (Romans 1 NKJV)
The power of God produces salvation for everyone who believes because in it the righteousness of God is revealed. Paul says salvation and righteousness are from the power of God. Power in how most translations render δύναμις. I prefer ability in this case. So with respect to the Gospel, salvation is God's ability to save everyone who believes and righteousness is the grounds or reason for doing so.
Another word which is sometimes rendered as power is ἐξουσία which means power as in authority. When speaking of the Gospel, the most relevant use of ἐξουσία is in the Prologue of John's Gospel:
But as many as received him, to them gave he power, ἐξουσίαν, to become the sons of God, even to them that believe on his name (John 1:12 KJV)
Both ability and authority are essential to becoming children of God.
1In comparing Romans and John, one would say John places focus on the authority God has given to man whereas Paul places focus on God's ability to save. Paul describes these in terms of salvation; John expresses this in terms becoming children of God. John opens with an explicit statement of how man must respond yet leaves the reader with two questions (1) believe what? (2) how do I exercise my authority? Paul begins with an explicit statement of God's ability and he also leaves the reader with two questions (1) the just shall live by faith...faith in what? (2) what must I do to show this faith?
Paul's opening statement has no explicit mention of authority, but the just shall live by faith implies they have been given the authority to do so. The significant term is, δίκαιος, just, which is the root of δικαιοσύνη, righteousness. Etymologically, the true authority of those who are just is from God's righteousness.
Paul and John describe the same thing in different ways and it is possible to over simplify their message. "Salvation" is seen in both individual and a group terms. Prior to Christ one who was Jewish believed they were saved because they were included in the covenants God made with the ancestors. The understanding of the end of times was from the group perspective, not from the individual's point of reference. That is, the individual understood salvation from being a member of God's people.
Neither Paul or John changed this. They simply placed the point of reference on the individual. God's people are still saved but the individual has assurance they are included because they are God's children on the basis of rebirth (John) or new creation (Paul) which comes from correct faith (Paul) or correct believing (John). Essentially, salvation and righteousness express both perspectives of salvation. The individual only is made δίκαιος, just by the δικαιοσύνη, righteousness of God. All who are just will be σωθήσῃ, saved (see below). They will experience σωτηρίαν, salvation (or deliverance) by God because God made them His family.
Romans 10:9-10 & Romans 1:16-17
After the opening statement σωτηρίαν and δικαιοσύνη are not put together until Chapter 10:
9 that if you confess with your mouth the Lord Jesus and believe in your heart that God has raised Him from the dead, you will be saved, σωθήσῃ. 10 For with the heart one believes unto righteousness, δικαιοσύνην, and with the mouth confession is made unto salvation, σωτηρίαν. (Romans 10)
In addition to the two terms in question, the second passage, like the first, uses a third closely related term. This time it is σωθήσῃ, saved, which is the root of σωτήρ from which σωτηρίαν, salvation is derived.
There are other changes to the first passage. Whereas Paul ended the first passage with δίκαιος the root of δικαιοσύνην, he begins the second with σωθήσῃ, the word related to σωτηρίαν. Also the order of σωτηρίαν and δικαιοσύνη is reversed. In comparing the arrangements of 1:16-17 and 10:9-10 they display a chiastic sequence with the words with δίκαιος and σωθήσῃ in the center:
Salvation (1:16) the Gospel of Christ is the power of God to salvation
Righteousness (1:17) for in it the righteousness of God is revealed from faith to faith
Just (1:17) the just shall live by faith
Saved (10:9) if you confess...and believe...you will be saved
Righteousness (10:10) for with the heart one believes unto righteousness
Salvation (10:10) with the mouth confession is made unto salvation
In addition to the chiastic sequence both both parts begin with mention of Jesus and God:
1:16 - Gospel of Christ...power of God
10:9 - "Lord Jesus"...God raised Him from the dead
Therefore, the two can be combined into a single thought which was interrupted after the quote from Habakkuk leaving the reader with questions which are finally answered in Chapter 10:
First Part: 1:16 For I am not ashamed of the gospel of Christ, for it is the power of God to salvation, σωτηρίαν, for everyone who believes, for the Jew first and also for the Greek. 1:17 For in it the righteousness, δικαιοσύνη, of God is revealed from faith for faith, as it is written, “The just, δίκαιος, shall live by faith.”
Two questions: What must I put my faith in? How do I show this faith?
Second Part: 10:9 that if you confess with your mouth the Lord Jesus and believe in your heart that God has raised Him from the dead, you will be saved, σωθήσῃ. 10:10 For with the heart one believes unto righteousness, δικαιοσύνην, and with the mouth confession is made unto salvation, σωτηρίαν.
Romans 10:9 picks up where 1:17 ended. It answers questions inherent to the δίκαιος phrase: those with faith that God raised Jesus from the dead which is demonstrated by the verbal confession "Lord Jesus" will be saved, σωθήσῃ. 10:10 gives the reasons why this "works" - with the heart one believes unto righteousness and with the mouth confession is made unto salvation.
If 1:16-17 and 10:9-10 are a single thought, the explicit mention of God's power, or ability which is found in the first part is continued by implication in the second part. That is, one with the correct faith and Lord is given righteousness and salvation which must come from the power of God.
If the two parts are a single thought there is no reason to sever the connection between σωτηρίαν and δικαιοσύνη which is explicit in verses 1:16-17. Both salvation and righteousness are from the δύναμις of God: one refers to God's ability to save and the other to God's reason for using His ability to save. A further distinction is possible in that righteousness from the human perspective is the source of what makes the individual just. Since this is from God it is also the assurance of being included in the people who have salvation with God.
Addendum - Why Link Salvation and Righteousness
One might ask what reason underlies Paul's connection of salvation and righteousness. For example, John's Gospel describes God's initiative from love (3:16) seemingly making righteousness secondary (cf. John 16:8, 10). On the other hand Romans, despite having important statements about God's love (cf. Romans 5:5, 8), begins with God's power, specifically connecting salvation and righteousness not salvation and love, as one might expect.
The fundamental difference between John's Gospel and Paul's is the point reference. John presents the Gospel in the form of first hand information coming from Jesus to which he adds explanations such as is found in the Prologue. Despite receiving the Gospel first hand from Jesus (cf. Galatians 1:11-12), Paul explains and defends his Gospel as being grounded in the Old Testament from which he adds explanation:
1 Paul, a bondservant of Jesus Christ, called to be an apostle, separated to the gospel of God 2 which He promised before through His prophets in the Holy Scriptures (Romans 1)
As much of Romans is in the form of answering objections about the Gospel, and specifically Jewish objections as to the inclusion of Gentiles, I suspect Paul was determined to present and defend the Gospel on primarily a "legalistic" position.
In the LXX σωτηρίαν and δικαιοσύνη are found together in Psalm 40:10; 98:2; and 119:23, and Isaiah 51:5, 6, 8; 56:1; 59:17; 62:1, and 63:1. The passage in Isaiah 51 displays the same structure found in Romans:
My righteousness draweth quickly near; and my salvation also shall go forth, and in my arm shall the nations hope; the isles shall wait for me, and in my arm they shall hope. Lift up your eyes to the heavens, and look upon the earth the earth beneath; for the heaven is massed as smoke, and the earth shall wax as old like a cloke, and they that dwell in the earth shall die as those (things); but my salvation shall be for everlasting, and my righteousness shall not fail. (LXX-Isaiah 51:5-6)
Like Romans, the two words σωτηρίαν and δικαιοσύνη are repeated in inverse order. δικαιοσύνη then σωτηρίαν in 51:5 are σωτηρίαν and δικαιοσύνη in 51:6. Unlike Romans the pair begins in the order Romans ends. Other than Isaiah 51:6, the only other place I found σωτηρίαν and δικαιοσύνη in the order Paul uses first is Psalm 98:2:
2 The Lord made known his deliverance; before the nations he revealed his righteousness. 3 He remembered his mercy to Akob and his truth to the house of Israel. All the ends of the earth saw the deliverance of our God. (LXX-Psalm 97:2-3 NETS)
The term translated as nations is ἐθνῶν, the word Paul uses for Gentiles. From Paul's perspective the Psalm reads, The Lord made known his deliverance; before the Gentiles he revealed his righteousness. No doubt Paul saw his own call to apostleship present in the Psalm.
It appears Paul's decision to begin with σωτηρίαν and δικαιοσύνη was taken from their connection in this Psalm and the decision of reversed repetition was taken from Isaiah. By doing so he echoes, or makes an allusion to Isaiah despite beginning with the Psalm:
I am not ashamed of the Gospel of Christ...The Lord made known his deliverance; before the Gentiles he revealed his righteousness...and so forth
Finally, while Paul's use of δίκαιος is from Habakkuk, the connection between δίκαιος and σωθήσῃ is from Zechariah:
Rejoice greatly, O daughter Sion! Proclaim, O daughter Ierousalem! Behold, your king comes to you, just and salvific is he, meek and riding on a beast of burden and a young foal. (LXX-Zechariah 9:9 NETS)
1. Paul also speaks of believers being members of God's family which he describes as adoption (cf. Romans 8:15, 23, Galatians 4:5, Ephesians 1:5)
2. This does not mean Paul denies love as God's initiative (see Ephesians 2:4). It is simply making the observation in Romans Paul gives God's righteousness rather love as grounds for acting to save.
3. Translation from R.R. Ottley, The Book of Isaiah According to the Septuagint (Codex Alexandrinus), Cambridge University Press, 1909, p. 269