Summarize this article:

Walking away from a confrontation with someone who hates you does not mean that you are admitting defeat or accepting the reality that a perfect solution cannot be reached. Instead, it demonstrates that you have complete control over the situation by not engaging in their negativity.  Haters never offer constructive criticism or positive feedback, only criticism. If you already know this then there is absolutely no reason to have a dialogue with them. Listening to a hater will only ruin your day. All of the negativity will do is hurt your feelings, make you angry, or incite you to participate in the pessimism. A hater will get under your skin whenever they can. It is important to stay calm when this happens so that a physical confrontation does not occur. Stay level headed and understand that a hater does not deserve a response from you, especially in retaliation.  By staying calm and not responding to their attacks you demonstrate that their opinions and destructive comments are not valued by you. Over time, this should reveal to them that attacking your character is a waste of time. Remaining calm will conserve your energy for more positive pursuits. Do not waste your time or energy on a situation that does not deserve your attention. Since a hater will seek out negativity and place undue criticism on you don’t submit to their turpitude. They will want you to explain your mistakes in front of others to reveal your imperfect self. In response, never justify, explain, or apologize to them until they approach you in a respectful and sensible manner.  Value your self-worth and don’t degrade yourself to their level unless they treat you with the respect you deserve as a person. In this case, it is probably unwise to reply with a smart comment or to try and play it off as nothing important because it gives them an opportunity to turn the tables again on you for not addressing their accusation to begin with. Let them embarrass themselves without help. This might be difficult to do if you work with this person or have to deal with them on a daily basis, but you can take steps to limit your exposure to them. By limiting face time, it will reduce how many times a day you have to take evasive action or submit yourself to their hate.  Simple changes to your daily schedule will throw the timing off just enough to help you avoid the hater. Try taking your breaks at work as different times; come home later in the day or leave earlier in the morning if the hater is your neighbor or roommate; or stay in different rooms of the house during holidays if the hater is an extended family member. In each of these cases, if you come into contact with the hater then either politely excuse yourself and exit the area or try to keep things neutral and unemotional if you must interact with them. Avoid bringing up topics that provide an opportunity for the hater to become negative towards you.

Summary:
Walk away. Stay calm. Do not offer an apology. Limit your interaction with them.