Romans 10:18-21 meaning

In verse 18, Paul is referencing a psalm of David in the Old Testament (Psalm 19:4). In this psalm, David explains that God communicates who He is constantly through all of creation, that the heavens continue to proclaim His glory every day and night. 

This is the answer to the rhetorical question Paul is asking in verse 19: But I say, surely they have never heard, have they? (v 18). 

The answer is clear, indeed they have (v 18). 

If they didn't hear the prophets, indeed they have heard God speaking through all He created. The Jewish people have heard this message of righteousness through faith (Romans 9:30-33). God has always been telling it to them, "Their voice has gone out into all the earth, And their words to the ends of the world" (v 18).

The Romans 10:18 translation of Psalm 119:4 differs slightly from the translation of the psalm itself:

"Their line has gone out through all the earth,
And their utterances to the end of the world."
(Psalm 119:4)

The pronoun "Their" refers back to Psalm 119:1, which says "The heavens are telling of the glory of God." The heavenly voices that speak to us of God and His creation reach everywhere, even to the "end of the world." The "line" that is spoken of refers to a plumbline, which is a tool builders use to ensure that their building is straight. In a similar manner, God's creation shows to us His ways, and leads us to know how to line up with, or live, in harmony with His ways. 

Even if someone has not heard audible words to explain God and His ways, God's creation gives them ample knowledge to know Him, so that all humans are responsible to respond to the knowledge they have (Romans 1:19-20). 

Paul quotes Moses (Deuteronomy 32:21) to give another example of a proclamation of the salvation of the Gentiles. Moses promised the Israelites that God would choose a non-nation (Gentiles, those who are not Jews) that will anger them and make them jealous, thereby leading the Jewish people to desire the gospel: But I say, surely Israel did not know, did they? First Moses says, "I will make you jealous by that which is not a nation, By a nation without understanding will I anger you" (v 19). 

Paul makes this point about jealousy again in Romans 11:11, 14. In Chapter 9, Paul made it clear that Jesus would be a stumbling block for the Jewish people and that Gentiles would come to righteousness through faith, the same way that the Jewish people must come to righteousness (Romans 9:30-33). 

In verse 20, Paul quotes the Old Testament prophet Isaiah (Isaiah 65:1), who also predicted that this would happen: And Isaiah is very bold and says, "I was found by those who did not seek Me, I became manifest to those who did not ask for Me."

God foreshadowed the Gentiles finding the righteousness of faith using numerous examples of individual Gentiles coming to faith in the Old Testament, such as Ruth, Namaan, and the widow of Sidon. In fact, Jesus referenced the latter two examples in an illustration of the rejection He got from his hometown of Nazareth, and it made the Nazarene Jews so mad they wanted to kill Him (Luke 4:16-30).

Verse 21 affirms that God has been telling the Jews the same gospel message that Paul reiterates throughout this letter. But as for Israel He says, "All the day long I have stretched out My hands to a disobedient and obstinate people." 

God has invited them, yet they refuse to listen. Paul uses quotes from the Old Testament prophet Isaiah to show the audience that this refusal is not something new. This same point was made by Stephen right before some Jewish priests and rabbis stoned him for his message:

"You stiff-necked and uncircumcised in heart and ears! You always resist the Holy Spirit; as your fathers did, so do you. Which of the prophets did your fathers not persecute? And they killed those who foretold the coming of the Just One, of whom you now have become the betrayers and murderers, who have received the law by the direction of angels and have not kept it."
(Acts 7:51-53)

Paul was in attendance for this event, holding the coats of some of Stephen's murderers (Acts 7:58).

Much of this letter to the believers in Rome is in response to the competing Jewish "authorities" who slandered Paul's gospel message (Romans 3:8). It is likely that this letter to the Roman believers is, in part, intended to support Aquila and Priscilla, who were fellow Jews who preached the gospel with Paul in Greece, and are now returned to Rome where they have started a church in their home (Romans 16:3-5; Acts 18:2, 18, 26). 

In chapters 1-8, Paul addressed the accusations and slanders against his gospel message, showing his Jewish competitors why and how Jesus abolished the law. 

In chapters 9 and 10, Paul has been addressing the larger question of why these Jews and many other Jews, because of their desire to pursue righteousness through the law, see faith in Jesus as a stumbling block (9:30-33). 

Paul has relentlessly insisted that righteousness comes only by grace through faith, for the Jews and the Gentiles. We gain righteousness in God's sight and are justified by Him simply through believing; we receive that free gift and receive it wholly by faith (Romans 3:22-24, 5:15, 18). Then we experience righteousness when we walk by faith, believing what we know to be true, confessing/thinking those truths, then living them out.