It's difficult, you see. First, the text:
Now we know that whatever the law says it speaks to those who are under the law, so that every mouth may be stopped, and the whole world may be held accountable to God.
In the context, it is very difficult to understand what that sentence is getting at. The 'easiest' reading, is if it assumes that everyone is under the law. If so, then that makes the second half makes sense, because obviously the whole world would then be accountable to God.
Having said that, this 'easy' reading is actually also very difficult because it means that you have to imagine Paul using the phrase 'under the law' to include everyone in the whole world. This is a difficult feat, especially in the context of Romans.
So, which is right? Are those 'under the law' just the Jews, or everyone?
Currently, I lean towards an idea that Jeremy Nicholas Wales has proposed. The previous chapter (2) of Romans spent a fair portion of its time establishing the idea that even those not under law were accountable to God for their actions. This part of chapter 3 is concerned with rebutting possible Jewish protestations that they ought not to be liable to judgement for their actions.
Once you've put those two together, you've covered the world. Jews are liable, non-Jews are liable. That pretty much covers it.
Hence, the point of v19 is to say that now the final piece in the judgement puzzle has been placed (ie, the Jews) it's clear that the whole world fits into the picture.
What do you think?