Few people would deny that both the Old and New Testaments contain texts decrying sexual diversity and even sexuality itself, despite its being the only method by which the commandment to "be fruitful and multiply" can be fulfilled. As with other issues, the modern theologian may interpret diverse statements in Scripture in such a way that, to reconcile them with each other, an absolute view of them, as might have been common in the past, is altered, accommodated to changing times and increased knowledge about the universe and about man.
From the New Testament, however, there is a seldom-cited text in which the question of sexual diversity, I believe, is answered more definitively than elsewhere in Scripture. Responding to a question about marriage from the disciples, Jesus said,
All men cannot receive this saying, save they to whom it is given. For there are some eunuchs, which were so born from their mother's womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven's sake. He that is able to receive it, let him receive it. (Matthew 19: 11–12 (KJV))
Some have construed this text as a form of abuse against gay people. I believe it is the opposite. The Greek word ευνούχος [eunuch] referred to anyone not of "normal" sexuality, not merely the present-day popular notion of men serving as harem guards. To me, the most reasonable interpretation is that Jesus meant diverse sexuality can be inherited ("born from their mother's womb"), environmentally rooted (those "made eunuchs of [by] men," though this could also refer to those deliberately deprived of their sexuality), or chosen by the individual. The last is placed in a spiritual context, but given the overall implications of the text, I believe it can mean a personal choice for different reasons.
The reason that few people cite this text is possibly that the use of the term eunuch makes them uncomfortable or is offensive. Many also point to the fact that in the Gospels, Jesus did not overtly discuss the topic of homosexuality or specifically condemn it.
Traditionally, sexuality has not been viewed as a nature-versus-nurture issue by the Church and faith traditions, but many Christians nowadays have developed more liberal views and beliefs regarding gender and sexuality based on interpretations like the one above. Ultimately, though, when answering this question, it is up to you to decide what you believe on this issue and how you interpret the relevant Biblical passages.