Gender is first of all an ambiguous term referring to a set of characteristics commonly associated with masculinity and femininity.
Gender is a social construct because its meaning is dependent on another gender--i.e., other people. If you were the only women in a universe of men, you would not be a woman--you would be a freak. If only women existed, the concept of gender would not exist.
I'm assuming you have a vagina, or at least that's what you've indicated in your post. From that we can theorize that you have exactly two functioning ovaries, though you might not. You probably have a uterus. Your breasts are probably larger than than the average male. The hormones in your blood probably fit within the average female ranges. Your voice is probably softer and higher pitched than mine (I'm male). You probably aren't as strong as me. You probably have similar tastes in music and TV shows as my girlfriend. You probably menstruate every 28 days. You probably want to have sex with men exclusively. You probably like pink. You don't play or like sports. You like to gossip in the bathroom. You like flowers.
Notice that all of these things are assumptions, stereotypes. You could fit any one of those descriptions without fitting any of the others. Now you can ask the question--are you still a woman if you have no uterus? What if your estrogen levels are low? What if you have a 36 day cycle. What if you don't like pink? What if you're not a good mother? Are you still a woman?
The definition of gender floats on a local average of a population's opinions of what gender should be. Personally, I also reject the idea that sex can be determined genetically. Chromosomes do not determine sex. You can be an XX and still be infertile, have a deep voice, play football and have abnormal hormone levels. There are XX males with Y chromosome material mixed in with an X. All sorts of in-depth genetic analysis could be applied to sort these things out, but the end result will always be comparison to some socially defined checklist of things males and females should have. There's the obvious stuff to look for like genes encoding for genitals--but is that where you draw the line? If I have a penis but no sperm, am I male?