Is it possible to stay white belt and continue to learn as normal until black belt level without taking belt advancement because not really interested in belt but in skill for self defense
Whilst it is of course possible and I agree with much of @Wigwam's answer.
I think it is important to look at the benefits of the gradings themselves:
dealing with pressure - you will be put under pressure at a grading, which of course is scary the first few times, but as you get used to this pressure you deal with it better, allowing you to deal with other high pressure situations better (which may include self defence situations). Of course the black belt grading will be of a higher pressure - at least with the previous gradings 'under your belt' it won't be such a step up.
seeing where you really are - am i half way to black belt? what aspects of my training are not as strong as they should be? Gradings answer those questions for you - they guide your training and allow you to know on a regular basis which aspects you need to work on.
respect - knowing what a grading involves and how hard people must have worked to pass them, will allow you to respect those senior colour belts (regardless of how arrogant they may be), it will also earn you the respect of your fellow students almost without question. (If a white belt questions something they will more likely be dismissed out of turn than a higher grade).
Martial arts are a journey and it is up to you the road you take - but having seen the positive impact on people the grading system has had I would always suggest trying it.
Is it possible? Well, we can send a man to the moon and bring him back alive, so it stands to reason that it is possible for you to stay at white until you're ready to test for black belt. Some styles do this as a matter of pracitice anyway - Aikido is notable for this.
But is it recommended? That's not so easy to answer:
Understand that the purpose of a colored belt is primarily for the benefit of the instructor. If not financial (as would be the case in a McDojang), it is for convenience. There are no pedagogical advantages, since belts bestow neither knowledge, nor wisdom, nor capability. But they can help an instructor divide up a class so that those who are more skilled can work on more advanced techniques than the more junior classmates. If your instructor has few students, this may not be an issue, and may be easier for a student to be granted the favor to lay low and remain a white belt longer. If your instructor is not your primary instructor (for example, there are many assistants, particularly those who may hail from another school), they may not know your true capability, and you may be assumed more junior than you are.
Thinking other ways, if your school competes, having colored belts can also help distinguish your capability for purposes of competition. You can be seen as a cheat if your skill level is of an advanced brown belt (with your black belt test the following day) competing in the white belt division. If you, or your school, does not compete, this may not apply to you.
My advice is to not worry about the belt. If you test, you test. If you don't, you don't. Just go with the flow.
There's a pretty big contradiction in your question.
You are not interested in belts, but you want to have a black belt. So which is it?
If you aren't interested in belts you shouldn't be interested in the black one either. You could find a school/martial art that works without belt systems or you could not care about your belt at all. Personally, belt doesn't really matter to me, but the challenges that come with the grade exams give you an opportunity to test yourself to your limit, to challenge yourself, to be creative and to be heckin' proud of yourself.
Also, you should know that a belt doesn't represent skill. All the TKD black belts I've met (this ranges from first to 8th dan) told me that when you reach your black belt it simply means that you can really start learning Taekwondo. Plus, being a black belt doesn't mean you can defend yourself more than someone who doesn't have a black belt. I like to think of the belt as a representation of my skill in comparison to my previous skill.
I believe it will depend upon the school. In the program that I took, skills and patterns only advanced upon graduation to higher belt/stripe levels. Consequently, if you remain a white belt, you would never learn any new patterns or skill used to achieve them.
Lots of solid points covered in existing answers, to which I'll add another aspect...
If you stay at white belt while those around you are grading and working their way towards black belt, it can create a certain "turbulence" in the school. Say another visiting student has been graded to some higher belt, but they visit your dojo or come across you at a cross-dojo training session. They'll see your white belt and seek to engage at a reasonable, supportive level to both challenge and encourage a white belt, but all that's based on misleading information.
You can dance around and they can humour you and gently poke a few techniques out and you can feel like you're mixing it up with someone of a higher grade, but sparring a X belt as a white belt is not the same as sparring a X belt as a fellow X belt or as their senior - having to prove yourself. Being visibly a senior is also about feeling the pressure to play your role in challenging and encouraging those who should be junior to you, if you weren't dicking around avoiding gradings.
What happens if you break our your not-at-all-white-belt moves, and surprise them? Well, it's not really fair to them - they're probably going gently on you and trying to make sure you're comfortable enough to learn effectively, developing trust in the practice/sparring environment, and you're exploiting that. They might feel they just can't get stuck in as it'd be over-the-top against a white belt, and you may misinterpret their reticence as some skill-based dominance. Not healthy for anyone.
Or, they might feel they have to defend the status of their rank, and really get stuck in to try to put you in your place. I certainly wouldn't have let a "X" belt get the better of me when I was a "X+1" belt, and while I'd cut a white belt a lot more slack, there's a limit. People may think you're being disrespectful, which could end badly for you. And if it does, they shouldn't go home thinking "damn - how did I end up having to beat up a white belt today? - I feel like crap". Other students may see them having to get stuck into you, and blame them for being over-the-top without realising the extent to which you were pushing them. To some extent, battling for dominance between the ranks is normal, but it should be done without misrepresentation. Or, if they can't control you they may wonder how a white belt - which represents someone walking in with insignificant training in the art - can best them, and doubt the system, their own ability, the examination process etc.. Other junior students may watch you dominate them and wonder how a white belt can overcome an X belt and doubt the system. None of that's fair or honest.
I can understand not liking gradings. And it's very comfortable being a "X"-belt-in-white-belt-clothing. But taken too far it's ultimately dishonest and disrespectful to those around you, including your instructor (who should step in to stop this before it goes on too long).