I did everything on my own regarding my B.Sc. thesis. My supervisor did nothing. I want to publish it without his name

The basic rule is that you can publish your own work without limitation. You can also correct previous versions, such as drafts, or even work submitted for class or a degree.

However, there are two considerations.

The first, is that in some fields, questions can be raised about "contributions". In a chemistry lab, that is grant funded and has a lot of participants and a PI who wrote and manages the grant, it is considered proper to include many (all) of those participants, often as co-author. If you are in such a field, and depended on such things, then you need to consider the contributions that made your work possible, even if the contributions were indirect.

The second consideration is purely political, but not ethical or legal. An advisor who likes you and your work can do you a lot of good in the future, especially if you want to aim for an academic career. In such cases either an acknowledgement (even if unearned) of an advisor, or even including him/her as a co-author, might just do yourself some good in the future. An adversarial relationship with an advisor, especially when you need them for letters and such is a career killer. Having a faculty member, advisor or not, who dislikes you, whether validly or not, won't help you and could hurt.

But the advisor needs to weigh in on whether s/he wants to be a co-author. It is improper to include someone who hasn't agreed. You don't need permission for an acknowledgement, however.

My approach would be to tell the advisor that you want to publish an updated version of the work. Ask for advice in doing this. You don't need to offer co-authorship, but if it is requested, my advice would be to submit. If nothing more, the name, assuming the person has a good reputation generally, will add a bit of weight to the paper, since you are a novice at this. It can be an introduction to the bigger world.

If the advisor agrees that it can/should be published, you can ask if s/he wants to participate in improving it or if you should just go on your own. The answer you get should be instructive.

One danger here is that some people will try to take it over. Before you start on improvements, make sure that you know how authorship will work. Do this at the beginning, and keep a record of the interchange so that misunderstandings don't occur later. This is one sort of transaction where (saved) emails can be a good thing.

If you get advice not to publish it you will need to explore the reasons. It may be that the work isn't "ripe" enough for publication, even if it was fine for the degree.

I’d tread lightly. If the advisor truly did not contribute or read it then you can try to broach the topic very diplomatically. Say something like: I want to get this project published and I think I have some valuable results. I’d really LOVE you to get involved with the project so we can submit it to a good venue. If not I completely understand and would like a chance at submitting this work myself.

If the advisor was truly uninvolved they won’t care enough to bother. In addition they may offer valuable contributions still in terms of polishing the writing (I don’t know you but I’ve never met anyone who was able to write a good paper alone on their first attempt). They can also help pick a venue, guide the literature review etc. if they haven’t already done do.

I will also advise that contributing to a paper can come in many forms. Somehow I can’t believe that the advisor contributed literally nothing of value to the paper.