This sounds like it would be in an action movie. "He is counting the days until he can get his retribution."
If it relates to a calendar, would "it counts the days" also be correct?
Yes. If "der Computer" is doing the counting (for example one of those automated displays showing how many more days until Christmas), then it would be "Er zählt die Tage"...and the correct English translation would be "It counts the days."