First, I'm Dr. Jackie, an interpersonal communication professor and relationship expert. I want to help you. Second, I am so sorry you are going through this. It must hurt a lot to have your gf so hurt.
Unfortunately it sounds like she is really afraid of being deported. And because of what she interpreted as a threat by you, she likely thinks that if you have another fight, you will call INS or some other agency to report her illegal status. And that fear is extremely real for her. So even if you say you are sorry and beg forgiveness, I am sure her fear greatly outweighs other emotions. So the first thing for you to do is accept that this is where her mind is right now. She is not going to just suddenly change her mind/feelings.
However, the good news is that you can always establish/rebuild trust. But the key thing to remember is that it will take time. When someone is hurt, whether it's intentional or unintentional, the cause of the pain does not just die one day. It takes a lot of trust-building work on the part of the person who hurt the other one. So now you have to ask yourself what you can do to rebuild trust.
Through the research of some prominent interpersonal communication scholars, we know that trust is built on mutual vulnerability. That is, when both people have reached a certain threshold of trust, they both disclose what their fears are, what makes each vulnerable, etc. In other words, sharing deep, dark secrets that only a few people if that know is a true measure of trust--if you have been imprisoned in the past or in a psychiatric facility or whatever the crime or situation was, it was pretty bad to the point that you don't want anyone else to know. However, in a close relationship, sharing such intimate, private things is what serves to draw two people together. The reason for this is the VULNERABILITY. That is, if the other person really wanted to, s/he could disclose the personal information shared and really make life miserable for the other person. If you want examples, pick up any tabloid at the check-out at your local grocery store. Hollywood couples are NOTORIOUS for this--because they have been hurt/betrayed, they seek to cause the other person pain. The easiest way to do this is usually to reveal intimate details about the other person to the world.
So in essence, think about this. If the way to HURT someone is to reveal the OTHER PERSON'S "dirty laundry" so to speak, then the opposite must be true also. That is, the way to REBUILD trust is to start showing the other person YOU can be vulnerable. In other words, open up more intimate details about your life/past to her. Let her see that you CAN BE TRUSTED because you are trustworthy and that you are opening up to her about YOUR intimate secrets. The more you do this over time, if she still does love you, she is going to learn to trust you again.
I hope this makes sense. If you want, we can set up a phone or Skype session in order to talk in real time. I'll send you the form as soon as I send this. Please let me know how you would like to proceed.
Take good care,