Going back to academia after quitting my Ph.D. and taking a year off? I am a Ph.D. dropout with a one year of career gap. I want to return to academia, but now  I have realized that it was perhaps the biggest mistake to take a gap after quitting my Ph.D. The purpose of taking a gap was to work on my communication skills and get a  new perspective on my work and my career in science. This could have been done within a period of 2-3 months but it became one year long because of my own recklessness. I allowed myself to distract from my goal and I started working on my hobby i.e. Painting. I lost myself there and completely lost track of my time. I was really enjoying everything about this new venture. I took it as a challenge to become better at my skills in painting. Eventually, I became the best in my group and got lots of appreciation. But then, I lost my interest while realizing that there was no further challenge and all there was left is to work and work while crafting your art which was never going to be perfect. It made me realize that what I actually love is to take the new challenges and work on myself. To be an artist was never a career option for me, all I wanted was to learn the skills. That's it! 

While counting my achievement in science, I  realized that it was an irresponsible thing to take a gap in doing something that has nothing to do with science. Earlier I had a scholarship and a very competitive mind that has led me to get a Ph.D. position in one of the best labs in the country. Now I don't have the scholarship. I want to go back to my world of research where I was accelerating exponentially. I want to tackle the challenges of science with a new perspective and better skills. I want to be productive more than earlier and ready to take all the tough challenges. 
But I feel embarrassed about my past mistake. Because of this, I come across as someone who is not sincere about her career. But I can't help it because the truth is that I did behave insincerely by losing track of my time.  I should have been conscious enough to realize the consequences. Now, what do I do? The past has gone and the vision of the present is crying for a better future. How do I make things better like before?
This could have been done within a period of 2-3 months but it became
  one year long because of my own recklessness.


I did not notice any "recklessness" in your post -- instead, it sounds like you have been very proactive in exploring for yourself what you want to do in life. Taking a year off may have been the best decision a year ago. Now you know more than you did then; don't fault yourself for what you did not know.


  I started working on my hobby i.e. Painting.
  I lost myself there and completely lost track of my time. I was really
  enjoying everything about this new venture.


Your enjoyment is important, too; not just your career. It sounds amazing to take a year off and work on a hobby. In fact, many career academics wish they could have done this when they were younger, because it is much harder to find time for such things the more work and research you get involved in. Remember that you did enjoy this, and enjoyment can be rare and valuable.


  But then, I lost my
  interest...It made me realize that what I actually
  love is to take the new challenges and work on myself. To be an artist
  was never a career option for me, all I wanted was to learn the
  skills.


Try to think of it as a positive realization, not a negative one. Many people spend a lot of time (decades even) before they realize they want to do something different. Learning about what you want to do helps you see the future more clearly.


  While counting my achievement in science, I realized that it was an irresponsible thing to take a gap in doing something that has nothing to do with science.


Why is it irresponsible? You may have given up your scholarship, but you learned a lot about yourself in the process. I am not sure you could have done this any other way.


  I want to go back to my world of
  research where I was accelerating exponentially.


You can do this. Don't give up on yourself and what you really want.


  But I feel embarrassed about my past mistake.


Why are you embarrassed? Was it truly a mistake if you learned a lot both about your hobby and about yourself?


  I come across as someone who is not sincere about her career.


You do not come across that way to me, at least. However, you do seem to be very upset and beat up on yourself.


  Now, what do I do? The past has gone and the vision of the present is crying for a better future. How do I make things better like before?


The first step is to understand that you have a lot to be proud of. You may not have the scholarship now, but you earned it before, and you are still the same person. You are highly capable and have spent a year progressing in a different hobby, which you are also capable in.

The second step is to get advice from academics who know you better about what is the best past forward to get back into good positions. It may be that you can simply apply again; I don't think taking a year off is necessarily a problem. Or it may be that it would be better to get some more experience first, such as applying for some industry experience, or getting a masters' degree.