I have read the 11 posted answers so far, and I think if I experienced this a few months into my first job, the advice provided might not have helped me. So, here are a few things to consider:
First, some companies operate this way: dysfunctional and terrible to some people, like you, but exciting and engaging to the people that participate. You do not have to like it, or participate. However, you also need a job and it isn't clear how much you should try to "conform" to a company like this vs. seek another job (which could be worse, as you probably have already imagined).
There are 2 things to address, first is handling your well-being, and the second is understanding how to handle your career.
I all the sudden get headaches almost everyday and sharp stomach pains
every single day. The job isn't stressful, but I realized after this
incident, it may be the people around me.
If you can, seek professional counseling to help you manage your stress. Ideally, talk in detail with counselor or coach to get perspective on how to personally manage your stress and symptoms. I had a job that gave me headaches for a full six months, but it was worth it to me to overcome the challenge. In your case, it sounds unlikely that you would want to tolerate your work environment "as is" but you should never feel helpless that you can't feel better, even when you are in a bad situation. Please seek help, if not professional, find a religious or other support group to help manage your stress.
I understand he was drunk and I could just let it go
Letting it go is not really an option. You will be anxious if you do not in some way address this. The company culture suppresses handling emotionally difficult issues. Some comments make this sound like supervisors will "handle this" or they will in some way empathize with you, but they have told you to ignore it, so you can assume they are ignoring it also.
I had a manager yell at me once, in front of my manager even. Neither one of them addressed it with their boss. When I finally had a conversation with their boss (by chance on a business trip), he was shocked and embarrassed that it happened.
So try to talk to your manager. Make that person clearly aware of what happened. Ask the manager if something can be done to prevent a reoccurrence. If your manager is not open to directly addressing the issue, let it rest - it probably is not wise to make demands or try to change things. If the manager wants to address the issue and asks what you need, be prepared with a list of outcomes like: assurance that at the next company party, someone will either limit his alcohol consumption, a conversation with the offending party about his behavior and a promise not to repeat it, an apology to you in front of the manager, etc.
If you manager is the type to tell you that you are "overreacting" or something, you are probably not a good fit for the company. Since most companies really don't have problems or cultures like this, it is likely you can find another company that suites you better. If your manager is not helpful, simply thank them for their time, tell them you will be fine and can figure it out. You just wanted to get their perspective. A person like that could hurt your situation. It is not worth the risk.
It is probably best you address this after a few days have passed, so that people know you are serious and this is not a reflex reaction to what may be "normal" for the company. As a member of the company, even a new one, you do have some influence in how the company culture is molded going forward. This is especially true if you talk to a counselor and they are able to affirm your concerns and provide more detailed and personalized guidance.
My question is, in my case, what would you do?
Seek counseling - starting your career after university is stressful, and you are very stressed. Learning to cope with stress is a valuable, life-long skill.
That said, this seems excessive and terrible for you, and probably you are not a good fit for the company. Try to talk to a manager or supervisor, and try to be clear about what happened. If that fails, consider that you should probably find a place where you fit better. Then take comfort that you can find a better place in life while having to endure this part of it.
If you focus and work on this, you will be better for it when it is done. And don't despair, you sound articulate, reasonable and intelligent. The best response to inappropriate behavior is to compose yourself and to be appropriate nonetheless.