Does divorce occur to the second wife if the father swore that if she greeted his mother with "Salam," she would be divorced, and then she went on to greet her and speak with her?

Divorce occurs if the father, by his oath, intended to prevent his wife from speaking to your mother. The consideration in an oath is the intention of the swearer, and intentions vary between general and specific. Ibn Taymiyyah holds that a conditional divorce intended for threat, urging, prevention, or emphasis does not take effect; rather, an expiation for breaking an oath (kaffarat yamin) is required.

Therefore, the matter must be presented to a Sharia court or to trustworthy scholars to ascertain the father's intention in his oath.