What to do when wrongfully accused of cheating? For a couple of days ago I had an in class online quiz consisting of multiple choices. I had read and prepared for the quiz and got a 80%, not too well but not bad either. 

Several days after the exam I got a email from my professor saying that two different persons had come to him and told I was observed cheating by searching the web for answers. I met up with the professor and he blamed me straight away for cheating. I never talked with him before but he sided immediately with the other side rather than me, even though he had no "past impression" of me. He tried to scare me to admit to something I did not do while I repeatedly told him I did not do it. He made me very uncomfortable, saying I should drop the class and so on... 

He was being very childish and immature, I told him it was unfair to take someone’s word over mine when there was no prior relationship, he just started working here. He kept threatening me and I told him I would never admit to something I did not do and it is wrong to accuse someone based on a rumor, this means that I could potentially wrongfully accuse someone of cheating and they would be in trouble. 

I am very scared now and I suffer from social anxiety, I have been scared to show up to class as I feel the professor is out after getting me. After this incident, he has also given me very bad grades, borderline failing me, even though I am a student with a 3.5 GPA and very hard working. Also he has been talking about me "cheating" to other students, started a rumor about me that has spread among several classmates, which makes me very uncomfortable being around people. 

I have gotten several reports from classmates that he has shared this private matter with them. Whatever I do in class now is not good enough; he keeps on punishing me with bad grades and gave me a zero on the quizzes, making the class hard to even pass. I have no clue what to do in such a situation, and as mentioned before I suffer from heavy social anxiety and I am scared of going to class. 

We had a quiz with the same format as the other one shortly after with same amount of questions from a different chapter, I had to finish that quiz while he was looking at me and I got the same exact result as the previous one. I thought that would be proof enough, but he told me that he believes the other person more than me, without giving any reason for why. 

I don't want to be labeled for something I am not or punished for something I have not done. Is he allowed to talk to other students about such a private thing? What about my rights? What can I do?
First, you should check and read the entire academic dishonesty statement at your university. This should give you an understanding on how the university normally treats these types of issues. Actually being proactive on contacting an office of academic integrity or student conduct, depending on its name at your university, can help you understand what the standard route for a professor to take for academic integrity.

While it is appropriate for a student to come to a faculty member to talk about another student's cheating, it seems to be a red flag for your instructor to ask others about your cheating by name.  A vague announcement about seeing that people are not looking at their own papers or looking at their phones is appropriate, but actively singling a student out for their academic misconduct could violate one's privacy in a case of academic dishonesty. This is something you should check with your university though.

Another person who may be of help is the undergraduate chair of the department in which this is occurring, but it seems in the comments you have tried.