If you need emergency leave for a death or imminent death in the family or some other crises that obviously requires your immediate attention, then yes, the Navy will be accommodating. In fact, if your family ever needs to contact you ASAP and arrange leave, then they can go through their local Red Cross who will help make everything happen. There are even times when they will arrange priority transport on a Space-A aircraft if one can be found.
Keep in mind that an “emergency” is something an ordinary and reasonable person would agree requires your immediate attention, not what you might consider to me an emergency, such as your wife is leaving you or your sister is about to become a porno star!
Hope this answers your question?!?!