"I did not go to church today."
Translation:Hoy no fui a la iglesia.
In English, there's a big difference between "I did not go to church today" and "I did not go to the church today". The latter is very specific, the former could be referring to any church.
DL didn't accept "Hoy no fui a iglesia". Perhaps Spanish speakers would never say that?
You are right, we wouldn't say that. Although I could say something like "hoy no fui a clase(s)" or "hoy no fui a casa", meaning my class(es) and my house, respectively; instead of "hoy no fui a la clase (la casa)", which could mean any class or house, not just mine.
So in short, you should always put the definite article, unless talking specifically about your class, your house, or some proper noun like the name of a place.