Paul paints two very different pictures in Romans 10: one that explores the undergirding of acceptance of the gospel, and one that examines the reasons for Israel's rejection of it.
What does acceptance of the gospel mean? Paul outlines two criteria - belief in the heart and profession of the mouth (vs. 9 - worth quoting in its entirety here: "Because if you confess with your lips that Jesus Christ is Lord and believe in your heart that God raised him from the dead, you will be saved.") There is thus both an interior and an exterior component to saving faith. There must be earnest, genuine belief and an outward, public witness. This is true of Jew and Gentile alike, says Paul, quoting the scripture: "For everyone who calls on the name of the Lord will be saved." (vs. 13)
Of course, to be able to respond to the gospel (to call upon the Lord), one must first hear and believe the good news. And for that to happen, a proclaimer must be sent.
In Israel's case, Paul asks, is the problem that they haven't heard? Or that they didn't understand? No, he answers these objections in vs. 18 and 19. Israel has both heard and understood. The problem - the reason for their rejection - is a spiritual matter. They are jealous for their status as God's chosen people, not willing to welcome Gentiles into the family (vs. 19-20). They are, in the words of Isaiah, "a disobedient and contrary people" (vs. 21).
But don't count Israel out just yet - Romans 11 will bring a look ahead!