The Mysteries of a Woman

We all been there...

Over the past couple of weeks, my better half seems to be drifting away. Now she is the sort of woman that is strong, independent, feisty and of course - NEVER wrong!!
She is great as a person, we have been together on and off for 3 years and I am so in love with her. But whenever there is an issue that we do not agree on, we fight for weeks. We are both very extremely stubborn, even over tiny trivial things. I believe part of it is that she thinks I am much more intelligent than she is and so she exaggerates her efforts to be an equal within the relationship.

Should I just agree with her, apologise and allow her to be right about everything?

or Do I stand my ground when I believe I am right?

Example: 

We argued the weekend and she said she did not want to see me on Tuesday. I text her Tuesday evening asking about her day etc. and got no response. I then went out for a few drinks with my friends. Later that night she complained via text that I did not tell her I was going out and therefore I do not tell her 'anything'. She also complained that I should have seen her before going out. Of course my point is that she had already told me she did not want to see me and also if she had text me back then we would have had a conversation and I would have told her I was going out and possibly been able to arrange to see her.

Any one have any views on this? Who is right? Me? Her? Neither?