I don't think you need to resort to hormonal or neural explanations. Staring has social meaning. The meaning of staring varies across cultures and contexts. In some contexts it is normal (e.g., staring at a presenter, staring at the person you are talking to, staring at a sales assistant). In these contexts, staring has meaning such as indicating interest or indicating an accepted attempt at engaging the person's attention.
In other settings staring can violate social norms and may be interpreted as an aggressive or annoying action by others. In some cases, staring can make people feel self-conscious or objectified. In other cases, a stranger staring at them can be experienced by the person being looked at as an intimidating action.
You can imagine a train of reasoning that might arise when you stare at a women long enough for her to notice and for you to notice that she has noticed. She could be thinking that she doesn't want to be stared at and that social norms say that she should not have to be stared at. Yet, you continue to stare at her. At the very least she might find this disrespectful. In some contexts, it might induce concern that if you are willing to break that social convention, you might be likely to break others, which give rise to fear that you might be violent. In other contexts it might just be interpreted as a sexual advance, which they don't want to reciprocate.
Anyway, the main point is that staring has complex social meaning, and thus should be understood within the social context including prevailing norms about staring in general.
It seems that the academic Rosemarie Garland-Thomson has written quite a bit about the meaning of staring from a humanities perspective.
See for example, her 2006 article
This article considers how staring informs the ways we know each other
and the world around us. Staring, a complex, nuanced, and
meaning-laden social interaction, can take many forms: arrested,
separated, or engaged. It is an intense encounter which is sometimes a
random, idiosyncratic confrontation and at other times a highly
structured social ritual driven by the collective impulse to look.
This article argues that staring often defines the relationship
between disabled and nondisabled individuals. More important, however,
it seeks to redefine this relationship by imagining, perhaps
counterintuitively, the object of the stare as determining the
structure and outcome of the staring engagement. The staree may take
charge of the encounter, in other words, using various strategies to
both mediate and transform discomforting interaction into an
unexpected opportunity for mutual transformation.
Or her 2005 article
Staring is a vivid form of human communication. Part of our enormous
communal vocabulary of the eyes, staring is a particularly emphatic
way of expressing our response to others. A more forceful and
sustained form of looking than glancing, glimpsing, scanning,
contemplating, surveying, gazing, and other forms of casual or
normative looking, staring starkly registers intense interest and
endows it with meaning. That interest ranges widely in form--from
domination, adoration, curiosity, surprise, allegiance, disgust,
wonder, befuddlement, openness, hostility, to reverence. The stare is
a highly charged interpersonal encounter that we snap up in a variety
of contexts to put a sharp point on what we mean, think, or want.
Staring is a way of strongly reacting to another; it bespeaks
involvement. It is the human response to novelty, to the unexpected.
As such, staring is an embodied and relational visual exchange that
carries complex cultural and historical meanings. Like sex and eating,
staring is drenched with significances, scrupulously regulated, and
intricately ritualized. Civility, for example, has always strictly
governed staring and prescribed what we do with our eyes in social
encounters. In American culture, the one thing everyone knows about
staring is that your mother told you not to do it. Both furtive and
compelling, staring is imagined as a formidable interchange and is a
source of vivid narrative within in the Western cultural archive.
She even has a book called Staring: How We Look. Oxford University Press, 2009. She talks about the book in this short YouTube clip.
- “Ways of Staring.” Journal of Visual Culture 5, 2 (Summer 2006): 165-184. PDF
- “Staring at the Other.” DSQ: Disability Studies Quarterly 25, 4 (Fall 2005). FULL TEXT