I want to use Bayes' theorem to calculate the probability that a totally hypothetical religion H is true considering the fact in light of two independent pieces of evidence. I'll make up values for the example, but let's say the prior probability of H being true $P(H) = 0.01$ (let's say one of 100 religions which includes H must be true).
For the first piece of evidence, a hidden skip code in H's ancient scriptures indicates that the ancients had supernatural access to knowledge of future events. If H really were true, we would expect about an 80% chance that some kind of code like that would exist. If H were not true though, there would only be a 10% chance that a code like that would randomly be present anyway. So I'll say $P(E1|H) = 0.8$ and $P(E1|H') = 0.1$.
For the second piece of evidence, archeologists discovered ruins which contradict with a claim in H's scriptures. If H were not true, we would expect a 50% chance of finding some such ruins. And there's only a 1% chance that the archeologists could be wrong or that it is theologically acceptable for this contradiction to exist. So I'll say $P(E2|H) = 0.01$ and $P(E2|H') = 0.5$.
So then I have an equation for Bayes' theorem considering these two separate observations like this:
$$P(H|E1, E2) = \frac{P(E1|H) * P(E2|H) * P(H)}{P(E1|H) * P(E2|H) * P(H) + P(E1|H’) * P(E2|H’) * P(H’)} $$
$$P(H|E1, E2) = \frac{0.8 * 0.01 * 0.01}{0.8 * 0.01 * 0.01 + 0.1 * 0.5 * 0.99} $$
$$P(H|E1, E2) = 0.0016 $$
Is this math okay? Or is there a problem calculating it like this, and if so what would be the correct approach?
You've got the correct approach, you are applying Bayes theorem correctly for the numbers in your example.