ANSWER: Regarding a sex change the Bible answer is "no," but this has to be indirectly derived from Scripture. To be honest, there is no text that says, "Thou shalt not alter thy sex." However, it is true that the Old Testament did not approve of men making themselves eunuchs (making their gender neutral in a sense), which strips them of the ability to sire children by removing (at least) their testicles (Deuteronomy 23:1).
Of course, eunuchs could still serve the Lord (Isaiah 56:3 - 6, Acts 8:27 - 39). Many of them, since they were slaves, had little or no choice about their sterilization in such a gruesome manner. Deuteronomy 23 seems to indicate that people should not voluntarily choose to mutilate their sex organs.
A fundamental limitation of modern procedures for changing our gender is that the organs implanted or created are not functional: A man who tries to change and become a woman will not be able to give birth, nor will a woman who becomes a man be able to sire children via intercourse. They are merely trophies or symbols, not a functional reality. Therefore, someone who feels he is a woman trapped in a man's body, or vice versa, really visit someone specialized in psychological counseling, such as the reparative therapy of Doctor Nicolosi, before he or she would surgically mutilate himself or herself.
Gender is not just a matter of the reproductive organs. Its imprint is on every cell through the XX or XY chromosome, including in the brain. As a result, although someone can take hormonal drugs and undergo surgery, he or she cannot really change his or her sex at a fundamental level. The mind-set imprinted in a developing fetus' brain by the presence of testosterone, for example, is not removable decades later long after birth by taking estrogen.
It is much better for people to learn to accept who and what they are physically rather than trying to change it surgically, since the results are so crudely limited compared to what would theoretically have to be removed and / or altered in order to transform someone's gender.
The innately different personalities of women and men have not transformed over the centuries. Despite all of modern Western society's confusion about gender roles, etc., the traditional sex-based division of labor speaks to something intrinsic to the human condition and to the way God made women and men genetically. For secular evidence that men and women are innately different in their personalities, you would want to read George Gilder's Men and Marriage, which may be the most influential yet intellectual anti-feminist book published in the past generation.
The differences we see between men and women are not mainly created by society and the ways little boys and girls are raised by their parents and teachers, but reflect biologically driven (which was created by God) realities. By accepting the teaching of Scripture, we merely accept what we could discover and reason from nature based upon anthropological and sociological studies, such as what Goldberg did in The Inevitability of Patriarchy and Why Men Rule.
There is something clear from Paul's statements condemning homosexuality in Romans 1. The ancient world was like the modern world in being confused about sex roles, but that is not the way of God.
"For this reason God gave them up to vile passions. For even their women exchanged the natural use for what is against nature. Likewise also the men, leaving the natural use of the woman, burned in their lust for one another, men with men committing what is shameful, and receiving in themselves the penalty of their error which was due" (Romans 1:26 - 27, NKJV)
In closing, the Bible does not approve of a sex change. Although we cannot cite a direct text on the matter, we can make a conclusion based on the evidence available.