It looks like this question has been through some edits, so to be clear, I am answering this version of the question, and for the purposes of the answer I will presume for the sake of argument that the perception of the OP of a sexist remark is a reasonable perception:
Suppose that you noticed a student saying a sexist remark to you, and the student is completely unaware of it. In the spirit of fighting sexism, I feel that I should maybe say something. But on the other hand, as a person on the higher end of the power spectrum, I feel that I would be bullying the student if I were calling him out on his sexist remarks (which he is probably not aware of). If I witness a student engaging in a sexist behavior, should I call him/her out? How can I do this without making the student feel intimidated? What would be the professional way to deal with such a situation? My question is: If a student commits a clear sexist behavior in front of you, will you call him/her out? If so, how will you approach it so that you don't appear patronizing/bullying? (I made a major edit, because many answers addressed my "background" story rather than my actual question. I do not see any satisfactory answer that fully addresses my question.)
This is one of the things I absolutely hate about “call-out culture”. There is a whole world of possible teaching that can occur in the massive gulf between “calling him out” and “doing nothing”. Many of the students we teach in university are barely out of adolescence, particularly if they are in the early years of their undergraduate degree. You are the mature adult in the room, so it is probably not a great idea to use that authority and power to belittle this young man, or “call him out” during class in a manner that belittles him. However, your instinct to do something is correct; we all want to teach young people to do better. It would be perfectly appropriate to pull this young man aside after class and politely give him your views on his remark, and invite him to have a think about whether that is an appropriate remark. Don't treat this as fighting sexism (fist in the air); treat it as teaching an immature young man and helping him to grow up. My experience is that young men --many only a few years out of adolescence-- are hungry for knowledge, including knowledge on appropriate professional standards of behaviour from older adults.
I have encountered a similar experience to this in my own teaching. In one of my tutorial classes on mathematics, I observed that two male students were drawing pornographic pictures and showing them to a female student they were friends with, and chuckling about it. Although the three of them were friends, she appeared to be uncomfortable with it, and was not a participant in the joke. During class I told the boys to settle down and get back to their work, in a way that stopped the behaviour but did not intimidate or belittle them. (My favourite line in these cases is: “Come on now; you’re not at the pub”.) Then after class I asked to speak to them and had a more serious chat with them in private. This was a stern but sympathetic conversation, trying to give guidance to young men that were not mature enough to see that their behaviour was making someone uncomfortable. Based on their mortified responses, and their concern for their female friend, I expect that they probably learned the lesson, and I did not see any behaviour like this formfrom them again. To me, this gives the best outcome – raise the matter as a teaching opportunity, but save any stern remarks for a private chat.