Does the egg sperm determine the gender of a human child?

The sperm. The egg is always an XX chromosome and the sperm can either by XX or XY if the sperm is XX it will be a girl if it is XY it will be a boy.

Edit: Actually, all girls have XX chromosomes, and all boys have XY chromosomes. But eggs and sperm are gametes with half the chromosomes of the parents. So all eggs have X chromosomes, and sperm can have either an X or a Y chromosome.

(So your intention in the answer was right, but just a small technical error.)