How to write an apology letter for a professor?
one could say... implying... I also commented on... I may have gone too far... do you evaluate my reaction as provocative?...
Perhaps you're being a bit lenient on yourself?
I think the only right thing to do is to apologize, in person, and own your awful (truly awful!) behavior. You didn't perhaps offend and disrespect her, you absolutely offended and disrespected her. In my opinion, you were completely out of line (and unprofessional to boot.) Regardless of the opinion of others (including my own), I think you need to deal head on with this problem you have of lashing out at others for (real or perceived) slights. Part of dealing with a problem behavior is dealing with the consequences.
People are more likely to forgive an offense if the offender has actually apologized and asked for forgiveness. Though she may continue to harbor some residual ill-will towards you (and deservedly so), it will show the community to whom she belongs that you at least tried to do the civilized thing after exhibiting such poor judgement. Also, it will likely stop her from continuing to bad-mouth you if you have apologized appropriately. It doesn't sound good to the listener if she complains bitterly about you, then ends with, "...and then he apologized appropriately."
Don't send a letter or an email. Have the courage to do it in person, and to allow her the opportunity to tell you what you probably need to hear. It might also help you to read about real apologies lest you continue in the vein which started this.
However, if you want her continued ill-will and bad-mouthing to follow you around the start of your career (it may or may not hurt you, you never know), do nothing, or better yet, continue to ignore the better angels of your nature, and tell people she had it coming to her.
OK, let's take this apart:
"One of the referees, a professor of mine with a history of verbal contentions between us, did me an overt injustice and despite all my efforts, attacked me with her unfair judgement." You may be short in your description, but she's a professor and so presumably knows her field. Have you had a moment of introspection in which you have considered that, just possibly, she may have been right? I mean, it's an exam and it's her job to find out how much you know. I have been to my fair share of exams in which the student simply did not know very much. On the face of it, the questions the students then tend to get from professors may sound harsh and are occasionally blunt ("I would have really liked it if you had known the answer to this -- it's second year undergraduate material"), but they're not factually wrong and certainly not "overt injustice" or "unfair judgment".
"I got angry and sent her an e-mail the other day, one could say explicitly offensive and disrespectful, implying her lack of knowledge and her biased criticisms." So you're saying that she's a professor but does not know very much, and on top of that is biased. Any reasonable outside observer would certainly say that (i) you're out of line here, (ii) that you're offensive, (iii) that you're almost certainly wrong.
"I also commented on her wasting too much time in social networks, rather than doing academic research." So you're saying here that she's not only unqualified (see above) but also not doing her job. If she's a professor, you're probably wrong on this -- first, how would you even evaluate how much she is or should be working; second, she may spend the occasional minute on social media (heck, I'm doing that too right now!), but she may be making up for it by working late at night or on weekends (of look, I'm doing that too!); and third, this is most definitely not your business.
"I heard she burst into tears after reading the e-mail, and has notified almost all other professors about it." So someone was completely and unnecessarily awful to her and she went to her peers and friends to talk about this.
"A couple of months ago, I defended my Master's thesis [...] I got angry and sent her an e-mail the other day." So you've been harboring a grudge for several weeks and finally lost your temper?
As others have pointed out in comments, this question may as well have been asked in the Interpersonal Skills forum, but since you're already here: Can you see how completely inappropriate your original email was? If you can't, then you definitely have a blind spot in your interactions with other people that you need to work on.
Now, about what to do: I suspect that there is really not all that much you can do, and there is really nobody other than yourself you can blame for this. But if you want to try: Be an upright person, take responsibility for having made a mistake, and go apologize in person and with sincere contriteness and regret. The last half-sentence is important.