I am looking for a word to use for when someone who cares about someone else wants to know how they are doing. The context this is in is that Person A injured themselves, and Person B, who cares for Person A, wants to know if they are O.K. (immediately after the injury).
Inquire after means to ask about someone's health and in my opinion carries the connotation of concern and interest. Although, you could inquire after with a false concern. But, then you are just being a jerk.
To inquire after: To ask about the health or condition of (someone).
(This is according to The Free Dictionary)