Romans 10:1-4 meaning

In Romans chapters 9, 10, and 11, Paul is addressing his concerns for the Israelites, God's chosen people. Paul himself is a Jew and wishes fervently that Israel would be saved. Brethren, he addresses them, my heart's desire and my prayer to God for them is for their (Israel's) salvation (v 1). 

He has written this letter to combat the slander of competing Jewish "authorities." He is likely writing in support of Aquila and Priscilla, fellow Jews from Rome who were co-workers in his ministry to the Gentiles, and are currently leading a church in Rome (Acts 18:2; Romans 16:3). These competing Jewish "authorities" have claimed that since Paul preached grace, he also preached that it glorified God to live a sinful lifestyle. The competing authorities argue that God actually wants us to live by the Old Testament law. Paul, however, says: For I testify about them that they have a zeal for God, but not in accordance with knowledge (v 2). The them he addresses is Israel (Romans 9:31). 

Paul denounced the characterization of his teaching of grace as granting permission to sin. He calls this characterization "slander" (Romans 3:8). He instead has insisted that the way God wants us to live is by faith in Him (Romans 9:30). 

Paul has proved that it has always been faith in God that brings righteousness (correct, harmonious living), going all the way back to Abraham, who lived hundreds of years before the law was written (Romans 4:1-4). But Israel sought righteousness apart from faith: For not knowing about God's righteousness and seeking to establish their own, they did not subject themselves to the righteousness of God (v 3). 

This has been the human problem from day one. Adam and Eve fell to the temptation to gain knowledge independent of God's ways; their error brought death into the world (Genesis 3:6). Satan tempted Jesus to do the same, but Jesus endured the trial (Matthew 4:1-10). Israel strayed from their vow to follow God's commands as contained in their covenant/treaty with God, again following their own ways (Acts 7:51-53). 

Israel sought to follow the law to obtain righteousness, but instead of following God's commands by faith, they rationalized their behavior (Matthew 23:2-5). The problem with the law is that it does not change the heart. An evil heart can always find a way to rationalize being right. But the law is no longer the agency that leads to righteousness, for Christ is the end of the law for righteousness to everyone who believes (v 4). The way to live righteously is not coming up with more and better rules. Rather, it is to follow Jesus Christ in faithful obedience, trusting in our hearts that His way is for our best. 

The question Paul is now responding to can be found in Romans 9:14, anticipating the competing Jewish "authorities" in Rome to claim that if Jesus's grace overrules the law, then God has broken His promises to Israel. Paul will insist that God's covenant/treaty with Israel still stands, although the people have broke their vows (Romans 11:25-27). 

Paul anticipates these dissenters to conclude, "So since the law is displaced, you're saying Israel is cast aside? Therefore, God is finished with it forever." If, as Paul teaches, Jesus ends the law for believers, the competing "authorities" argue that the promises of God to Israel must be set aside as well.

Quite to the contrary, Paul argues. Jesus has established true righteousness for everyone who believes. The Gentiles (the Roman believers) have righteousness through faith, and these competing Jewish "authorities" attacking them do not. Why? They're not seeking righteousness by faith. The stumbling stone (faith in Christ) trips them up (Romans 9:32).

Paul declares that the Jews have a zeal for God. Zeal (Greek: "zelos") means an intense enthusiasm or passion. But their zeal is dictated by their own construct of how to live. The Israelites' zeal has not led them to obey God, but rather to justify themselves. God's righteousness (living as God intended, living harmoniously with God and others) comes by trusting in God, leading to obedience to God from the heart. God gave the law to the Jews so that they would see their need to rely and trust in God (Galatians 3:24, 25). And as Paul stated in Romans 8:4, the law is fulfilled not by making rules but by walking by faith in the Spirit.

Paul has been more than clear throughout his letter to the Roman believers that life comes through faith, and living harmoniously as God intended (righteousness) comes through following the Holy Spirit by faith; we were given a new life, which was made possible by Christ's resurrection. Christ gives us a new life. The Israelites rejected Christ and His gospel. They have clung to the law, which grieves Paul, because it keeps them from justification in the sight of God and from conforming to the image of Christ.

In Galatians 2:21, Paul conclusively says that Christ died unnecessarily if the law brought righteousness. But it doesn't, it only brought more sin through man's inability to obey it (Romans 5:20). In verse 4, Paul writes that Christ is the end of the law for those who believe. The law is finished, it's of no use anymore to believers to lead us to righteousness because we have Christ now. To the extent the law reflects Christ, we can still benefit from it, because it points us to Christ (James 1:25).

Everyone who believes is declared righteous in the sight of God, whether Jew or Gentile (John 3:14-15). Through Christ, we can finally start living the way God designed us to. In the Old Testament, Abraham attained righteousness only by faith in the promises of God, and it's by faith in the death and resurrection of Jesus Christ now for believers in the New Testament. Then, when we live by faith, the law is fulfilled (Romans 8:4).