I became pregnant when I was already a Christian having a relationship with a Christian guy but we weren't married. It seems to me that his obligation is that he marries me. Does the Bible say he has to? And what if he doesn't and dates someone else? Even though our daughter was the result of sin I just can't believe the father doesn't have a responsibility. God hates broken families right?
The Bible doesn't say whether he has an obligation to marry you or not. Under the Jewish law, if the woman was a virgin and not engaged or married the man was obligated to pay a bride price to the father and marry the woman. The New Testament does not go into such details of law in this case.
Legally, the father may have a responsibility for support, but not necessarily for marriage. That would be something to discuss with a lawyer. Morally, some might say he has an obligation to marry you, but he may not be interested in what is morally right. That is something both of you should have considered before you got into this situation.
God does hate broken families, but since there was never a "family" established there isn't one to break. If you were never married then there is nothing to break. You and your daughter are a family, but he was never part of that. As much as he should be a part of that family, you can not force him to be, and if you were able to force him that would probably result in an unhappy marriage for all concerned.