A.T. Robertson Commentary Romans 3:19

"Now we know that what things soever the law saith, it speaketh to them that are under the law; that every mouth may be stopped, and all the world may be brought under the judgment of God:" — Romans 3:19 (ASV)

That every mouth may be stopped (ινα παν στομα φραγη). Purpose clause with ινα and second aorist passive subjunctive of φρασσω, old verb to fence in, to block up. See 2 Corinthians 11:10. Stopping mouths is a difficult business. See Tit 1:11 where Paul uses επιστομιζειν (to stop up the mouth) for the same idea. Paul seems here to be speaking directly to Jews (τοις εν τω νομω), the hardest to convince. With the previous proof on that point he covers the whole ground for he made the case against the Gentiles in 1:18-32.

May be brought under the judgement of God (υποδικος γενητα τω θεω). "That all the world (Jew as well as Gentile) may become (γενητα) answerable (υποδικος, old forensic word, here only in N.T.) to God (dative case τω θεω)." Every one is "liable to God," in God's court.