"As the Good Book Says" is a common trope where a character quotes or misquotes the bible (or other holy works). The character is often attempting to appeal to authority, in this case, a holy book.
And in fact, Tevye prefaces a few sayings not with "Is it not written in the holy book", but with "As the good book says". So your memory had the right concept, but the wording was slightly different. Here are some of his quotes...
Heal us, O Lord, and we shall be healed.
If you spit in the air, it lands in your face.
When a poor man eats a chicken, one of them is sick
good news will stay. And bad news will refuse to leave.
Each shall seek his own kind.