If a woman can produce gametes with x chromosomes, and a man can produce gametes with either an x or a y chromosome, and the resulting offspring will be made up of one gamete from each parent, does the father or the mother determine the gender of the baby?

In terms of results, it would generally e up to the male parent to determine whether it's a boy or a girl. If it where up to the mother, she would only have one way of choice, but the father, who hosts the "y" chromosome, provides possibility in more than just one associated gender. I hope this helps.