Can an air traffic controller refuse to give service to an aircraft based on moral grounds?
No.

The purpose of air traffic control is to prevent collisions and expidite and maintain an orderly flow of air traffic. An air traffic controller has no say in who gets to fly and who doesn't. Refusing a certain flight would be something that needs to be done at a political level, and certainly far away from an operational level.

Also remember that, except in special cases, air traffic controllers have no way of knowing who is onboard a certain aircraft. Essentially, all we see is a flight ID and a route. Air traffic controllers are not in contact with passenger lists at any time.

Furthermore, air traffic controllers do not have any physical way of preventing flights from taking place. See this slightly related answer: Can a pilot take off at his own discretion?