How to apologize sincerely for harsh words spoken to the most precious person you know?

In my unqualified opinion…

If you place someone in high regard, when you argue with them about anything at all, you will automatically become more aggressive and less patient, just because of the urgency and extra stress that is put on the situation, by the high regard you place on the person.

At least that is what I think about my self after thinking on how I have behaved at times.

What do you do?

Pull them to one side when there is a moment, and explain that you are sorry. You tell him/her that you don’t like being that way and that you are working on it, and repeat to them again how important they are to you and how you feel about them.

The other person will then try to brush it off, as if it is no big deal anymore, you then say it is a big deal to you, and repeat your self again with yet another apology.

It sounds like there are two things at play here. One is that you still feel badly. Two is that you are uncertain of the damage that your words made.
You stated that you two were good, but is that the case? Has the other person accepted your sincere apology and acting in a way that indicates that as being past it? If that is the case then do not keep bringing it up with that person. The work is on you now along with proving through time that you do not act disrespectful again.
If there still is indication that the damage is deeper, then sincerely ask for time to discuss it and carefully state your feelings and then juust listen to their side.
Ask a qualifying question of “are we good, can we more past this, is there any other concern.
It is key that if you agree that it is spoken through and agreed that it is a finished matter that it stay that way. Bringing up things from the past that you thought or agreed were over is not healthy.

The rest then is up to you to work on not acting in that way again. Soul search out why you may have acted like that. It is more than likely something from your past that you didn’t let go of or learn to dea with differently.
We are made up of our past experiences. Our future is made up of what we do with how we act in the present.

Sounds like your real problem may be that you are too nice. I agree with @poisonedantidote, you may feel she is too perfect but that would mean that you don’t find yourself good enough. Which may mean you try to please too much and aren’t honest with what you feel and with what you need. This is going to build into resentment and spill out once you feel you have reached your max. Forgive yourself because this is natural human behavior. However, knowing what you know now you should feel guilty if it happens again. No matter how sweet or caring she is, a really great relationship depends on honesty and being open with each other. Have a talk with her and explain that you need a relationship where you both are equal partners and for that too happen it will mean that you both need to voice your opinions and wants without being judged and that compromises will be needed.
If this still doesn’t work, write down what is bothering you. Read the first draft and then re-write anything you feel would be too harsh if it was written to you. Keep writing it till you point is clear and you are not attacking her for past misdeeds. Just write about how and why a certain thing may make you feel a certain way.

You’ve told her; from here on in you must show her.

Then you and she may want to learn how to deal with conflict without starting World War III. That will prevent a recurrence. This is a skill that all couples need to develop. No relationship runs smoothly all the time.

Do not do it again because you could lose her. Remember that, and try to forgive yourself by vowing not to let it happen again.

Actually, you might want to say what you wrote: I am feeling like these words are still in the air and I don’t like myself saying them to the most important person in my life. I admit I was very nervous and I spoke out loud and garish but it still bothers me. I do not want that to happen ever again.

Or something like that. Then do it.

The words are lingering for her too. She is wondering what you really think of her: what you usually say or what you said when you were angry.

Tell her openly and honestly that the words just rushed out of your mouth and were not really about her. Tell her it was pent-up anger from your past.

Then tell her good things every day for a while.

Unless you notice discomfort and unease coming from her, I suggest not bringing it up again. If you bring it up again, that will just make her re-live it . . . unnecessarily if it’s just “your side of the issue” that you’re dealing with.

Then, as others have said, don’t do it again. EVER. The best way to forgive yourself is to resolve that nothing like that will ever happen again and work every minute of every day to make sure that it doesn’t. When nothing even remotely close to what happened occurs in your relationship, she will forgive you as well.

You didn’t say whether this kind of thing has happened before with other people. If so (or if you do it again despite all your efforts), you’ll probably want to explore counseling. At that point, something deeper is going on, and you may not be ready for a relationship until it’s resolved.