Why does Indian Hindustani or Carnatic music not use chords in compositions? I haven't seen any composition that mentioned about chords. Were Indians not aware of chords?
There are some chords, but they're used in a bit of a different way than the Western classical tradition. It's a stylistic feature of these two genres owing to their origins in the late Medieval and early Renaissance eras.
They both highlight the raga (basically a melodic pattern) and tala (a rhythmic pattern), just like the respective color and talea of the isorhythmic tradition in Central Europe around the same time.
But whereas the isorhythmic tradition often emphasized the motet (and thus polyphony), the Hindustani and Carnatic traditions tend to emphasize a single voice (or vocal line) in a homophonic texture. Often this doesn't result in chords, but sometimes it does.
It's not that they didn't always think of chords, it's just that this is a stylistic trait. It's a bit like asking why Dickens never wrote an illustrated children's book. He could've, he just didn't. (At least I don't think he did.)
All traditions of a culture are inter-related. Indian philosophies have focused on the individual not collective. Music is a path for an individual to seek bliss or its variants. Every element of music then immediately takes on either an individual or supportive role. This gives every swara (simplistically a note) a status and character. This explains why creating a sound by combining multiple frequencies is missing. In its place, Indian music works with microtones between swaras.
I think it's because Indian classical music is primarily a vocal musical tradition. We have some very beautiful instruments too, but if you've noticed, they all try to approximate the nuances of vocal music.
Most historians agree that the beginnings of Indian classical music go back to the Vedas, the Sama Veda in particular. To explain a bit about the Vedas, they are the world's oldest extant literature. They come in the form of four vast collections of hymns that were orally transmitted from generation to generation.
It was forbidden to write the Vedas down, and yet, there was an unflinching commitment to preserving them at all costs without any changes in the intonation of even a single syllable. You see, the Vedas were considered to be literally a way for humans to communicate with the Gods, and it was thought that even a slight mistake in intonation would change the meaning of the communication and result in catastrophe.
Vedic scholars were, therefore, trained to ensure flawless recitation of the hymns taking care to use the prescribed notes for the prescribed syllables.
In the beginning, only three notes were used in the Sama Veda. Eventually, the number of notes increased to five and remained there for a while, before the sixth and seventh were finally added. So, the Sama Veda is what began this tradition of very disciplined and rigorous training in music, which eventually evolved into the very rich tradition of classical music we have in India today.
So, to go back to your question, Indian classical musicians have always taken their notes very seriously. The purity of the note is considered to be of the utmost importance, especially when you're practicing your music or if music is a spiritual pursuit for you.
I think this is the primary reason we don't use chords as a deliberate musical device.
Having said that, the tanpura is used as a background in Indian classical music, and it is usually tuned to either the 1st and the 5th or the 1st and the 4th, and played continuously in the background. This results in an inadvertent chord-like effect, but since these background notes are simply played over and over again without any deliberate thought given to them, they do not distract the musician from concentrating on the note he or she is focusing on.