In its phrasing, the question of whether biology or society has a stronger influence on the development of gender identity seems to be arguable. It does not fail on the assumption that either is the sole determinant of gender identity. In asking which is stronger, the implication is that any other determinant of gender identity is not part of the debate. Unfortunately, these same limitations promote the idea that these two positions are the only positions worthy of debate, and thus consideration. In that very construction, it becomes fairly evident that society plays the stronger role in defining behavior in terms of gender, imposing the division with little regard to the true determinants of identity - the manner in which individuals define themselves based not only upon an understanding of their own thoughts and feelings but also in the context of the language and culture in which they must express themselves.
The debate between nature and nurture is relatively blind to the struggle we all face in trying to define ourselves based on both who and what we are, especially in those instances where an individual experiences a conflict between who and what they understand themselves to be. Thus, a meaningful conclusion is not liable to be drawn from a debate on the influence of society vs. biology on the development of our gender identity. The debate itself does, however, introduce several essential considerations on each side. On the side of biology, there are the arguments of genetic inheritance and predisposition, hormonal developmental influences on genetic expression of anatomic and neurological sex, and cognitive function. On the side of society, there are the influences of behavioral reinforcement, the effects of abuse or neglect, and neurolinguistic programming.
To determine if biology has a stronger influence, it is first necessary to decide how significant we consider the foundations of genetics on the development of the mind itself to be. The problem here is the lack of a clear understanding of the relationship between genetics and neurology; we have to ask how much genetic predisposition defines us, and how universal that genetic predisposition is. Observation of other biological processes in human development tend to suggest that many of the traits we manifest are manifested in response to stimulation; what we do determines what we become most proficient at doing. In this sense, biology provides us with the capacity to think and act and so the question of what we becomes depends on the thoughts and actions we engage in. The uncertainty here lies in not knowing which thoughts and actions will leave the strongest impressions, for how long, and why.
The study of the brain will continue to open doors on such questions, but based on studies of genetic behavioral predisposition using twins, we certainly do have genetically based predispositions that determine what we are receptive to or not. A person might be more likely to respond to stimulus that would trigger distinctively masculine or feminine thoughts and behavior, but if subjected far more to the opposite stimulus then the opposite behaviour might emerge. Because human beings evolved to favor adaptability, we are not driven by instinct alone. Our intelligence is a product of our evolutionary need for the greater problem solving capability necessary for survival adaptability. That problem solving capacity plays as great a role in allowing us to define our identities. It is so powerful we are literally able to assume any identity within the scope of our experience. This "acting" ability is just another survival trait, one that allows us to be whoever we need to be in order to deal with each other.
When probing the topic of identity, we tend to be referring to the person an individual sees him or herself to be. The problem with identity is that it is an expression of who we are based on our understanding of our own selves. In most cases the definition of who we are includes and encompasses what we are, but because societies typically divide human characteristics into masculine and feminine, it is not uncommon for people to define themselves in accordance to a trait that is contrary to his or her physical sex. This is where the question begins to shift from a biological stance to a social stance. The people around us influence us consciously and unconsciously every moment in which we interact. People provide the stimulation that shapes our social behavior, men and women both. Our predisposition might determine which influences we are more responsive to, but it is the people around us that determine which influences we are exposed to, and if our response is deemed inappropriate, it is the people around us who will actively seek to reinforce the "normal" behavior.
Under these conditions, the formation of identity is an ongoing and internal battle in which we have to weigh our need to be true to ourselves against our desire for acceptance. In every case, circumstances favor individuals who are predisposed to their anatomic sex, and in most cases an child's identity is not strong enough to survive antagonistic social pressure to conform to expected standards when they are not predisposed to their anatomic sex. The thing that is missing from the nature vs. nurture debate is an understanding of the fact that while a person can choose to identify with either gender, the actual gender of an identity is based upon an understanding of what makes an identity true to that individual. It is ineffable. No one can tell you who you are. Only you can know, and only when you understand. Whatever that understanding is, that is what you are - because anything else would be lying to yourself.
Biology and society are the anvil and the hammer with which identity is forged. Biology provides the foundation against which all the pounding of society is played out, but while one provides all the options, and the other the opportunities to explore them, it is the experience that generates the understanding that ultimately defines our gender identity. At this point, it might seem that there can be no victor to this debate, if both sides play equal and opposite roles in the process under debate. There is, however, one other implicit assumption in the debate that has to be reasserted. The question asks which has the stronger influence on the development of gender identity, and if the anvil and hammer analogy is to be used, then the strength has to be assigned to the tool that has the active role. The strength of the entire world might underly the anvil, but by itself it provides no action.
So, in conclusion, society provides a stronger influence on the development of our gender identity than biology. However strong the role, however, it is not the deciding force or even the defining force. There is a temptation to say that it is simply the process that reveals the true nature of the person, but there is meaning in that statement only if it is understood that a person is not merely a body. A society that fails to understand that is ultimately an obstacle to the development of gender identity because it will consistently fail to accept people for who they are and deny them the freedom to express themselves honestly.