It doesn't matter! Your girlfriend is sure you have made a mistake, and nothing you can argue will change her mind on the matter. She is being stubborn, and until this problem can be fixed, you relationship is going nowhere fast.
By the way, there are not too many boyfriends skilled enough in the psychology of women who can get them to apologise first. If you are able to do that, then you are one of an elite group of guys, that's for sure.
So, let's get onto the part where you actually say sorry to your girlfriend.
The first way for a boyfriend to apologise to his girl is by way of admitting guilt and fault. In this version of the apology, you simply tell your girl that you unreservedly, totally and completely accept all blame for whatever it is you did wrong that has caused her emotional distress and anguish. Whether you are sincere about this or not, doesn't matter. As long as she believes that you mean it, then you have succeeded. You need to learn how to make an unreserved apology under all circumstances - whether you have done wrong or not. Guys, if you are too proud to be able to do this, then we have to move onto another form of apology that might work.
The second way of saying sorry to your woman is to just hint at it. Using this tactic, you turn the focus on the situation. You say for example, "Honey, I'm sorry for the way things turned out there," or "I'm so sorry that accident happened." In this way, you spare yourself the imagined indignity of saying that you, personally, are accountable for whatever happened.
But, by far and away the best and most rewarding way to say sorry to your girlfriend, is to give her a present. Yes, women are at their most forgiving when their men turn up at the door with a beautiful and fragrant bouquet of flowers. Most of the traditional gifts will (usually) work wonders. Using this method, you can even openly admit that you are sorry by jotting down a few humble words in a card that accompanies your gift. If you include tickets to the movies, then you are sure to be forgiven for whatever you did (or didn't) do. By laying on the presents, you may even score an apology from your woman. This is because you are softening her up by showing your love through your generosity of spirit. Not many women are able to maintain any kind of anger or grudges when their guy brings a peace offering.