Can songs be added to an iPhone library without using a computer or the iTunes store? I wish to use my iPhone without having a computer. 

However, the iTunes music store is not available in India. So, is there any way for me to directly add music to my iPhone Music app without using a computer? I do not want streaming solutions, or any 3rd party apps to store the music.

I couldn't find any way to do this, but thought I'll confirm here once.
You can’t do that without jailbreaking. Even with jailbreaking, the apps are so buggy or expensive that I won’t recommend anything.

You have to use iTunes to sync music to your iPhone because you don’t have the Music Store in India.

The second best thing you can do is use third party apps to store music.