Not exactly. I suspect that Paul meant the same thing David meant (big surprise?). Which was certainly not "each and every single person is only unrighteous". I might say "no one can assume they are righteous because they are jewish, or because they are circumcised or perform certain other rituals."
Sorry, this is one of the very least spinnable scriptures there are. To say babies are sinnless is to say that they don't have any need of grace. It would be to say that they are worthy of heaven when they are born up until some point where they start sinning.
Rom: 9 What then? are we better than they? No, in no wise: for we have before proved both Jews and Gentiles, that they are all under sin;
10 As it is written, There is none righteous, no, not one:
11 There is none that understandeth, there is none that seeketh after God.
12 They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.
13 Their throat is an open sepulchre; with their tongues they have used deceit; the poison of asps is under their lips:
14 Whose mouth is full of cursing and bitterness:
15 Their feet are swift to shed blood:
16 Destruction and misery are in their ways:
17 And the way of peace have they not known:
18 There is no fear of God before their eyes.
19 Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God.
20 Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin.