Do you have to confess Jesus with your mouth to be saved?
YES!!! That’s literally exactly what Romans 10 says. In case v. 9 was somehow unclear, v. 10 clarifies it. The belief leads to righteousness, but salvation comes of confessing with your mouth. That’s just what it says. The only thing for any faithful Christian to do here is shout a hearty “AMEN!”
Having done that, we need to be sure we understand what Paul actually meant, because he’s not a post-Great-Awakening American like we are, and in contemporary English we have some ways of using religious language that are–by biblical standards–a little odd. When we use the word “saved,” in a spiritual context, we pretty much always mean “going to heaven, as opposed to going to hell.” So you can say “The lifeguard saved me,” and we all know you mean that he pulled you out of the wave pool, but when you say “Jesus saved me,” we all think of heaven.
Our default meaning for “saved” is a biblical usage–you can see Paul use it in Ephesians 2:1-10–but it’s not the biblical default meaning. When you see the word “saved” in the Bible, you need to ask a few more questions, like “Saved from what?” In Romans in particular, we need to remember that by the time we encounter 10:9-10, we’re ten and a half chapters in. Let’s not forget what came before it–you did read what came before it, right? Paul has already told us what he means by “saved” here; he began the theme in ch. 1 when he told them he was ready to preach the gospel (1:15) to the faithful saints in Rome (1:7-8), explaining that he’s not ashamed because the gospel saves–there’s our word–those who believe (1:16). We tend to impose our default meaning on 1:16 — you believe and now you’re going to heaven, i.e., “saved” — but that’s not quite what it says. Paul is saying that the gospel saves those who already believe, and he makes that clear in the next verse by quoting Habakkuk 2:4, which is plainly talking about someone who’s already a just man being saved from physical death by his faith. (Think Daniel in the lion’s den, or the three Hebrew children in the fiery furnace — they are among the direct recipients of the promise in Habakkuk.)
Paul goes on to discuss how we’re all justly damned in 1:18-3:20, and then begins to discuss the possibility of attaining God’s righteousness apart from the Law, through faith in Christ (3:21-4:25), and then moves on in 5:1 to discuss our status once we’ve attained that righteousness, being justified by faith. Here’s where our word “saved” comes up again: “MUCH MORE then, having now been justified by His blood, we shall be SAVED from wrath through Him” (5:9). Being “saved” in Paul’s usage here is much more than being justified. Like Abraham, your faith is accounted for righteousness, and you are justified on that basis — the same thing 10:9-10 says: “for with the heart one believes unto righteousness.” But God has more — “much more!” — in store for you than that.
Paul goes on to discuss how we deal with sin in this life (according to 6:23 it’ll kill you). As he develops his argument, Paul makes it clear that while arranging your mental furniture correctly is important (ch. 6), it’s not enough to keep you out of sin (ch. 7). Actually living resurrection life in your dead/mortal body requires active intervention by the Spirit (8:10-11), and when we allow the Spirit to work through us, He intervenes not only in our deeds, but even in our prayers (8:26ff). And if our own mortal nature can’t separate us from God’s love, then nothing can (8:31-39). So “saved” in Paul’s usage here has nothing to do with going to heaven when you die, and everything to do with being delivered from physical death. To borrow a paraphrase from St. Schwarzenegger, “Come with me if you want to live.” God’s faithfulness to His people is nowhere better demonstrated than in His continuing pursuit of His people Israel, which is Paul’s subject in ch. 9-11, where our text appears.
In that context, Paul asserts that we receive righteousness through faith (as discussed in ch. 4, and which is what we mean in contemporary English by “saved”), but being delivered from God’s wrath on sin in this life requires more than that. Specifically, you need to be willing to speak up.
Full Contact Christianity 
“… saved FROM wrath” and death from sin, but also saved TO something when we accept “intervention from the Spirit.”
Will we be seeing a Part 2?
I hadn’t planned to — this post came of a terrible discussion thread where a bunch of otherwise conservative theologians were falling all over themselves to say that you’re not saved by confessing Christ with your mouth. These people who claim to believe the Bible and then can’t manage to say “Amen!” when someone quotes an actual verse…they make me nuts. This post models what I wish they’d been saying instead.
But your suggestion is a good one — might write a second half after all!