I believe you might be seeking the 2004 novel Wasteland of Flint by Thomas Harlan. It is the first book in the series "In the Time of the Sixth Sun". Additional information here.
In five short centuries, the mighty Empire of the M,xica, descendants of the ancient Aztecs, has spread out to conquer the earth. Now it is ready to leave the homeworld and set its sights on the star. But the universe it finds is a dangerous place filled with hidden powers. Humanity is only a minor space-faring species on the fringe of a ferociously political arena where ancient and enormous alien empires are engaged in millennia-old struggles for supremacy.
Now, on a desolate, barren world far from the heart of civilization, Gretchen Andersson, a young human xeno-archeologist is about to discover an awesome, long-buried secret that could alter the galactic balance of power forever.
Points that match:
Around 2005 or 2006 I read a novel
Published in 2004
• it was actually present-day, but where the Native Americans went West and conquered Eurasia;
Partial Check. This is set in the future. But in an alternative timeline in which the Aztecs rose to power, not Europe and Western civilizations.
•The title was something like "the flint age".
Check. Title contains word Flint
•The priest were the highest caste in society, followed by the military, and was very racially motivated. Only Native Americans could be priests, and it was rare for any other race to advance in the military.
Check. The Aztec priesthood controls society much as it did in historical times in our timeline.
•An Asian was the captain of the spaceship in the novel, and lamented on how he was already far ahead of his station in life.
Check. This book is set with the Aztecs being the primary culture followed closely by the Japanese. Navy Captain Hadeishi is in command of the ship.
•White people were the lowest caste, and there was some social commentary on white peoples addiction to smoking, which was considered "spiritual".
Check. Europeans/Whites are the lowest caste.
•There was a cat-person on the spaceship who did not like smoking. (I think it was an alien who was a bipedal intelligent feline, but it might have been a non-alien cat/human hybrid.)
Possible Check. Do not remember this character in the book. But, there are other alien species so may be included.