As a state no. As a society, maybe. such state would sooner or later clash with one of the monotheist civilizations. If it would exist in some remote parts of the world outside of main "happenings" it would fall behind both technologically and military and would then be wiped out/annexed at lesiure (being in remote area it would alos have low population and being outside trade routes also not wealthy).
Look at India.
In OTL, Menander of Bactria conquered a lot of Northwest India. He went farther than Alexander. Alexander had turned back at Beas, without reaching Sutlej. Menander held Mathura, on Yamuna closer to Agra than Delhi, so how far did he go?
In 2nd century BC, Scythians invaded Bactria and wiped out Greco-Bactrians. Indo-Greeks held out at first. But in 1st century BC, the Scythians invaded India, too. Last king of Greeks was overthrown by Scyths in Sialkot about 10 AD.
Christianity did spread East a bit after 1st...5th century. There were Malabar Christians in South India, and Nestorians in Central and East Asia. But they were never so numerous and prevalent as to wipe out polytheism.
Islam invaded in 12th century. But polytheistic Hindus did keep resisting. Rajput states in Himachal Pradesh, Nepal, Rajasthan fought back. They often acknowledged overlordship of Sultanate of Delhi and Great Moguls, but many of them resisted outright annexation. And many Hindus kept their polytheism under the persecuting government of Mohammedan rulers.
Yes, the hill princes were somewhat poor, isolated and backwards. Yet they adopted warfare with cavalry and firearms, and built forts.
Now make a PoD somewhere in 2nd century BC. Menander is a bit more successful.
The result being that the Sakas do not conquer and wipe out all Indo-Greek states. Kashmir and Himachal Pradesh stay Greek although the plains of Punjab fall.
A succession of Greek kings and republics then rule the area through 1st millennium AD. Far from Mediterranean. There are still Hindus, and Buddhism is common (since Menander, as per OTL), but the Olympian pantheon is still worshipped, Greek temples built, and books of Greek philosophy are written (alongside books in Greek on Mahayana philosophy).
In 13th century, Sultanate of Delhi invades. Again, the Greeks resist wiping out (as did Rajput Hindus in OTL).
With the result that in the end of 18th century, the British East India Company officials, coming from Europe with their education in classical Greek language, mythology and literature, go and find Greeks who (unlike the Balkan Greeks under Turkish yoke who have been Orthodox for 1400 years) do and always have observed classical Greek religion and literature and are counting towards, say, 500th Kashmiric Games.
What will the Philhellenes like Lord Byron do?