Hello and welcome to Beyond Blue. It's awesome when people want to help and support another person struggling with depression, especially partners. Many people are afraid of depression and have no idea what to do. I will offer some suggestions.
First of all, your GF's actions are very common. Many, many people push others away when they are having a crisis. The rationale is they do not want to hurt others, place unnecessary strain on others and generally stuff up someone else's life. I know, I've been there and I'm struggling not to do the same again now. What people cannot grasp when they are in this state is that those they push away feel more hurt by this rejection than by the strain of helping someone they love.
You would think it was natural to turn to a partner when in trouble of some sort, and in other circumstances that's what happens, but not with depression. The Black Dog is truly a mongrel.
Knowing as much as possible about depression is the best start. On the home page of BB click on Learn About Depression in the middle of the page. This will take you to all sorts of information. Some you can download, other booklets etc BB will send to you on request, no charge. If you click on Resources at the top of the page the drop down list will offer more information, some for family and friends. I recommend you get these as well. The better informed you are the more you can help.
Does your GF have any professional help? This is really important. If not, see if you can get her to see her GP. Depressed people do not necessarily have to take medication or see a psychologist or psychiatrist, although I suspect, from your description, she should be getting help from at least one of these options. What about her diet? Depression can result in under or over eating. Either way she needs to eat the good stuff. It depends of course on her living situation, whether she lives at home, with friends or alone.
Actions like these will demonstrate how much you care for your GF without constantly telling her so and giving her the opportunity to push you away. Similarly with asking if she is OK and other questions. So keep an eye on her moods and be ready to listen when and if she wants to talk. No judgmental statements such you should go for walk or should do this and that. Just be there, ask questions but ask gently and infrequently. Once she realises you will take things at her pace she will be more inclined to stay. It really becomes a trust thing.