‘For Moses writes that the man who does the righteousness which is of the law will live by it,'
Paul is satisfied that he has now paid enough attention to the situation of the Jews with regard to righteousness, and thus refers to it only briefly as ‘the righteousness which is of the Law'. His concentration is rather now on presenting the positive side of the Gospel. But he refers to the righteousness which is of the Law again in order to contrast it with the Gospel and in so doing brings out important aspects of it. Moses had written that ‘the man who does the righteousness which is of the law will live by it'. The reference is to Leviticus 18:5 where it says, ‘you will therefore keep my statutes and my judgments, which if a man do he will live in them'. This reference is used by Paul in Galatians 3:12 in order to evidence the fact that ‘the Law is not of faith'. It can hardly therefore have any other meaning than that here.
Here then ‘the righteousness which is of the Law' is defined as ‘keeping God's statutes and judgments', and this had very easily slipped from being a loving and grateful response to the God Who had redeemed them, which was what God had intended, to being in practise a determination to observe a highly detailed set of rules which they saw as explaining God's requirements. They got bogged down in the detail. And this was in the hope that they would thereby ‘fulfil the covenant' from their point of view, so that God would have to fulfil it from His. They saw the ultimate consequence of this as being that they would receive ‘life', and their perception of ‘if a man do he will live in them' was that it referred to the way in which a man could have eternal life (this verse was regularly cited in Jewish tradition). What Moses was, of course, meaning was that men could thereby enjoy fullness of life (he had no real conception of eternal life). But the two do equate in that ‘eternal life' in its earthly aspect (John 5:24; 1 John 5:11-13) is indeed fullness of life (John 10:10). In this, in the view of the Jews, lay the Jew's hope of final salvation.
Note the emphasis on ‘doing'. It appealed to those who believed in a righteousness resulting from works. But Moses was not thinking in those terms. He was concerned with what followed redemption, and was stressing the benefits of then obeying God, an emphasis with which Paul would have agreed. But the Jews misunderstood it and saw it as teaching that the way to eternal life was by doing the Law, that is, that doing the Law as an important part of the covenant would cause them to inherit the benefit of eternal life. It is this idea which Paul is seeking to counter.