The common phrase and some variations are:
As God is my judge, I’m not guilty!
As God is my witness, I’m not guilty!
God knows that I’m telling the truth. I’m not guilty!
This is a logical statement.
“Since God is my judge, [therefore] I’m not guilty.”
The two parts of this sentence have no real logical relationship to each other. You could use “since” in this way:
“Since God is my judge, this court/judge has no real authority over me.”
Here both parts of the sentence deal with the authority of a judge.
Arinker gave examples of how this normally said. I think it’s also somewhat archaic.
I’m not sure it has the meaning of since or because. I think it might be something like “God is acting as my witness.” I’m not sure the exact meaning is important. It’s not meant to be literal. It’s more like an oath.