Absolutely Yes, and Absolutely Not, with a definitely Maybe inserted in there, I think. (laughter)
This is not a trivial question, because much depends on the intention and perspective of the person asking the question. If you are looking for a white-bearded gentleman about 60 miles in a geospatial orbit, then you are very likely to be disappointed, and many Hindus would agree with you that THAT God does not exist.
Hindus approach the idea of Divinity by exploring the true nature of reality, the same underlying motive that drives modern Physics and Biology, but the Hindu seers worked to identify events and their causes down the chain of causation. There is something, but where does it come from? That underlying Truth (capital T) must be the closest that the western mind can come to the idea of “God”.
However, if you are looking for proof that there is an order to the Universe, then you will agree with those Hindu sages who looked for Rta (Sanskrit ऋत), the cosmic order, and then inferred that the order must be maintained through some processes and an organizing principle. We can see evidence that the Universe is not random, and that laws of physics seem to be followed. That’s your indication that there is a divine order.
If you are looking for physical proof, then just open your eyes and close them. What you see with open eyes is a Universe that seems to exist materially, even though scientific microscopes seem to find only probabilities and space between. A ball that retains all colors except red, but still you will call it a red ball. This is all a play of consciousness, and we say “Jagan Mithya, Brahm Satya”…the material world is an illusion, only the knowing Self is the Truth.
So when you close your eyes, YOU still exist. When you die, your body exists, but not YOU. And if YOU exist, then surely there is SOMEONE that exists (even if it’s just you). Recognizing the self, or the Self behind all subjective experiences is the recognition of God.
Words can be limiting in experiencing the Limitless. “God” is a small word, and that which has created everyone cannot be constrained into it. The vocabulary of spiritual experiences can be broader, and the subjective experience of causeless bliss is proof of divinity to all those that have tasted it.
For Hindus, all-seeing is the Seeing of God, and all Tasting is the tasting of God. Raso-vai-Sah (“He is taste”).
Other reading material:
Relevant videos: Swami Nikhil Anand Hinduism Q & A: Does God Really Exist
Contributor: Gaurav Rastogi
Location of this post (1h): https://www.hinduspeakers.org/ufaqs/does-god-really-exist/