How can you help someone who does not want to be helped? (Long question details).

My girlfriend has depression. Although it has never been formally diagnosed, I know this is clinical depression because I have studied psychology subjects as a part of my degree and have done extensive assignments on it. She does not see a psychologist and is not medicated.
For the last week or two, we have been in less contact than normal as another downward spiral has taken hold. We spoke recently, and her thoughts were scattered and illogical (and my logic annoyed her). She refuses what help I offer, says that nothing matters, is easily irritated, and wants to shut out the world.
I don’t think she is in any direct danger, but I don’t want her to stay in such a state for obvious reasons. It stresses our relationship, but I love her too much to leave.

So considering this, my question(s) are:
What things can I do to improve this situation, considering that I am the only one that can say or know anything at all, and professional help is out of the question?

In general, can you help someone who does not want to be helped? How? Not just in relation to my situation or mental health at all.