First of all, kudos to the OP for sharing his problem. Without a doubt, this the first step towards addressing the problem. Unfortunately, most of the answers do not seem to get to the "meat" of the problem. And the main question is:
Was the OP rude to the student and was the student on the right to complain about this behavior?
The answer to both questions is a big YES. When someone asks a question about a test quiz and gets an answer like "I do not want to tell you because I might commit suicide because of depression" the student (in his human imperfection), has the right to feel "scared" and "awkward". But even then, the student still made the right choice: He notified the university "authorities" (the main instructor of the course) about the problem he faced. This is what we suggest all students to do (in this SE forum) when they are dealing with a dangerous or threatening situation within the university campus. And although the danger was not real, the student felt that way, so he reported it. What would a teacher do, when one of his students told him when asked about a meeting, "I do not know if I can make the meeting tomorrow because I might be dead"? Exactly the same thing. Notify the university authorities. So first, I believe we must all agree that the student did the right call. Otherwise, the OP might not even have acknowledged the problem and how the student felt from this situation (which I am not sure if he totally gets is 100%). And BTW commenting on "I do not want people to be unhappy with me for trivial events like this in future", let me tell the OP that being "scared" by your university teacher is not a trivial event at all.
I am not a doctor and I cannot provide remote diagnoses. No one should do it especially online. But being a teacher is mostly about the students and not ourselves. We are still human, we are imperfect and we sometimes make irrational mistakes. But if something (disease, personal state, even grief) prevents us to do what is best for THEM, we should do whatever it takes (therapy, medication, personal leave) to protect them and be the best teachers we can be. Otherwise, we are not doing those young people justice.