The terminology, "the righteousness of God," appears 8 times in this section of Scripture. We must understand that there are two completely different kinds of righteousness - God's and man's. Man considers righteousness to be something that is generally ethical in nature, but God's righteousness is inherrant to His nature. We think of righteousness in terms of human acts, but God IS righteous in His nature. Many times in the Old Testament He is called "The LORD our Righteousness." But carnal man takes great pleasure in his own righteousness, to the point that he cannot accept a righteousness that comes from outside of himself - one in which he does not participate in. He takes pride in attaining a righteousness that he has earned.
On Romans 10:3, Charles Hodge has said, "The grand mistake of the Jews was about the method of justification. Ignorance on this point implied ignorance of the character of God, of the requirements of the law, and of themselves. Those who err essentially here, err fatally."
There are five fatal consequences for those who have a lack of understanding of the distinction of God's righteousness. They are:
1) They are satisfied with their own righteousness
2) The look down on other people
3) The resent Jesus and His gospel
4) They misunderstood and mishandled the law
5) They will not submit to God's righteousness.
Such are the fatal and damning consequences of not believing that Jesus Christ became sin so that we might become the righteousness of God in Him (2 Cor 5:21).