Yes, your employer can ask about your family emergency and does NOT have to let you leave work for it. The employer can legally make you choose between your job and your family (and employers effectively do this everyday, even in more subtle ways--such as requiring working parents to work schedules that conflict with child care obligations). There is no legal right to leave or miss work for a family emergency. So you could be terminated for this.
Generally, employers may be more likely to question if it's a valid excuse if family emergencies or other absences are a regular thing or if you're new. If you're a dedicated employee with the company for several years who never misses work, they may be more understanding. However, company policy can dictate the same rules for everyone regardless of work habits.
That being the case, the employer can inquire into it, set conditions on it, or warn you that this was a one-time exception.
There is one exception to this. If you have applied for and are using FMLA leave, the employer may be required to let you off up to a point. If you're facing ongoing family or personal medical issues, talk to your human resources department. They can advise you on what kind of documentation they would need.
In these cases, as long as you follow protocol, you cannot be fired for related absences. If you exceed the amount allowed by federal law, you could still be terminated. Keep in mind, FMLA also has to be approved before you start using it.
Some companies may also provide mental health days, but that's at their discretion. In general, even if you have valid reasons, your employer may ask follow-up questions before letting you go or after you come back.