He who regrets not hath lived not. This may be true, but it doesn't explain the entire reason why a human being can not possibly live without regret. Not only can a life completely free of regret exist, it is unhealthy to never regret anything.
Every person who ever walked the earth did something they were not proud of, or didn't do something the wish they had. Out of these two categories stems regret, though many people also attach regret to events they could not have controlled in any way regardless of their actions.
First, there is the lack of action resulting in regret. Perhaps you wish you'd done something that would have furthered your own position in the world, like taken a job offer for a fledgling company that turned into a Fortune 500. Alternatively, your lack of action may have caused harm to someone you love or caused their lack of advancement.
While regret is often not constructive in these cases, it is a normal human reaction and may spur you on to being more proactive in the future.
The other main cause of regret is directly causing harm to another person. In this case, if you feel no sense of regret for your actions it could be equated to the lack of a conscience, or the lack of a soul. Feeling indifferent to the pain of others, especially pain you yourself caused, is sociopathic behavior and should be closely examined to find the cause of the indifference.
Now, while humans cannot live without regret, they can live free from the shackles of regret. Feelings of regret can be a weight around the heart; they can be a heavy burden that a person carried with them throughout their lives. That is, they can be burdensome if they're allowed to be.
On the other hand, feelings of regret can be constructive in that they can be a reminder of what you did wrong and what you'd like to correct in yourself. History is the best teacher, and those feelings of regret keep the lesson alive in a person's mind.
While it is not possible to live without regret, it is very possible and healthy to get to a point where regret does not rule your life. Regretting is an important lesson and it's important not to be indifferent, but it's equally important to recognize that the event is in the past and there's absolutely nothing that can be done about it except to improve the future.
A thorough examination of one's feelings of regret can help weed out those that really were out of your control, and to find the lesson in those events that could have been changed had you had a given quality. These qualities may be confidence, assertiveness, compassion, self control, or any other such virtues. In such cases, regret becomes the guideline for acquiring virtues, and then the memory of the events may serve as an indicator of self-improvement that has taken place since the regretted events.