In Brave New World, why is it ironic that the Director of Hatcheries and Conditioning becomes a father?

The irony lies in his title and personality. Remember that this is a society where concepts of "birth" and "mothers" are obscene beyond anything we could imagine. Life is created artificially; children are "hatched" and "conditioned", rather than "born and raised". When we first meet the man in charge of this aspect of society, he is overconfident, arrogant to the extreme, and enjoys shocking those around him. This is evident in his lecture to the boys in the first few chapters, as well as his planned public humiliation of Bernard, which is of course interrupted by his own, much more shocking revelation.

Linda confronts him in the Fertilization room, where they first met. The farce has come full circle, as it also the Director who has caused his own downfall by "fertilizing" Linda. The whole of his life was built around avoiding this very scenario, and the casual ease with which he slept with women has now been cast in a different light. He has committed the ultimate sin in this society (although they certainly wouldn't phrase it in religious terms) and must now live out his days in shame and alienation.