I'm 100% sure that conferences can, and I'm 95% sure that the NCAA can. But I don't know for certain what their bylaws say about fines for on field violations by players.
I've never heard of an player being fined by the NCAA. If so I suspect the judge that ruled against players as employees would have taken that into consideration. Further I suspect there is no such bylaw as the NCAA is composed of member schools not student athletes. The NCAA didn't fine Sandusky nor Paterno they hit PSU with penalties. Games aren't vacated from a coach but from the member school which in turn impacts the coach's record as a consequence.
The purpose of the rule was to reduce the number of hits deemed to lead to head injuries. Before football had rules players lowered their heads as battering rams. It's a natural reaction. The NCAA is trying to change a culture that starts with the players on the field.
The rules committee needs to review the film of the calls in the off season and tweak the language and the guidelines for the refs. I'd like to see a clarification of the rule where the ball carrier drops his head preparing for contact and actually lowers his upper body and head into a situation which creates a foul, ala Tuitt. Tuitt was in position to make a tackle when the runner lowered his head and entire upper body causing Tuitt to get a Targeting foul when he had targeted the runners midsection before the runner reduced his profile. This is similar in basketball where a shooter "draws a penalty" by leaning into a defender who has jumped in the air to defend a shoot. Only in basketball the consequence isn't a potential head injury nor a game suspension.
Around the same time as Tuitt, I believe Redfield got a deserved Targeting call when he planted his feet and coiled his body and launched forward to the neck/chin area of the opponsing player. Our player but a clear violation nonetheless. He was headhunting.
I'm for making the game safer but the rule needs to be refined to pervent the Tuitt's trying to make a legal tackle from getting tossed because the runner's action was the actual cause of a foul hit not the defender's.