My argument that 'theo' is the root of both is correct. You found the Latin but not the Greek Etymology.
He provided both, and following the links, I find:
"
From Greek "contemplation, speculation", from ‘ "spectator," literally "one looking at a show", from *+‘ > ()‘."
Nothing to do with God, you note. In fact, the English word with which it plainly has a relationship is
theater.
Edited by Dr Adequate, : No reason given.