When an acquaintance reveals a sensitive medical problem to you, how involved should you get?

Someone you don’t know that well (like an acquaintance) tells you about their personal medical problem. They are going to get tests done and find out if it is going to be even more severe or not. Is it okay to ask them about it later on? Or should you wait for them to come to you? Also, is asking by email okay or should you wait till you meet them face to face next time?

I don’t want to seem apathetic by not asking them about it since I know what’s going on but I also don’t want to seem nosy (even though I am genuinely concerned). I understand that they willingly mentioned the problem to me but that could have just been a low point in their life and they just wanted to let it out? What’s the etiquette on this anyway?