In business environments today, both men and women are expected to shake hands and there is no rule about who should offer his or her hand first.
On a social occasion, however, it is generally up to the woman to offer her hand first. If in modesty she says nothing, or in Japanese tradition just bows, it would be considered rude. In–stead of offering her hand, she may choose to smile and say, “How do you do?”
If the man offers his hand to shake, however, she would be rude to ignore it; she must shake his hand. Two women meeting for the first time may either shake hands or just smile and say hello, but two men will invariably shake hands.