Does ATC have to maintain separation on diverging aircraft?
Yes, this is allowed, subject to certain conditions. See FAA Order 7110.65 (the ATC "bible"), section 5-5-7, for the full details.


  ...all other approved separation may be
  discontinued and passing or diverging separation applied when ... Aircraft are on opposite/reciprocal courses and you have observed that they have passed each other; or aircraft are on same or crossing courses/assigned radar vectors and one aircraft has crossed the projected course of the other, and the angular difference between their courses/assigned radar vectors is at least 15 degrees.


That's in the terminal environment; it's still possible in the enroute environment but a bit more restrictive due to the greater aircraft speeds and poorer radar accuracy.