A drew two black balls, so he must have the BBB or BBW box. Since he knows the color of the third ball, his box is labeled as one of those two possibilities and the third ball has the opposite color than what's on the label.

Similarly, B has the BBW or BWW box with one of the respective opposite labels. From the information so far, we can already rule out 2 of the 4 possibilities: (BBW,BBW) and (BBB,BWW), as the same label would otherwise have to be used for both.

C too drew two white balls, but makes no statement about his knowledge of the third ball. He must have the BWW or WWW box. In the following part I will assume that C not mentioning he knows the color of his third ball means he doesn't know it, else my solution would not be unambiguous.

If C had the BWW box, that leaves B with BBW and A with BBB. The corresponding labels would be A:(bbw), B:(bww), C:(www). However with the (www) label C would know the color of his third ball, so we rule it out.

Thus C has the WWW box, now let's look at the labels of boxes A and B, for which there are two possibilities (we previously ruled two of them out).

a) (BBW,BWW) labeled as (bbb,bbw). Thus C must be labeled (bww). In this case however C would know the color of his third ball, so we can rule it out yet again.
b) (BBB,BBW) labeled as (bbw,bww). Thus C must be labeled (bbb).

The last case fulfills all the assumptions made so far and is the conclusion I have come to:

A has (BBB) labeled (bbw)
B has (BBW) labeled (bww)
C has (WWW) labeled (bbb)

Therefore D can conclude without looking at his box or its label that he must have the BWW box with the www label.