Tomlin didn't like Harbaugh saying the tougher team won after their last meeting.
IMHO, it doesn't matter if Tomlin doesn't like it. Then punch him out after the game and after the press has left the building. Or walk across the field and go into his locker room and hash it out in Harbaugh's office. Or fire barbs back in the press.
Many of the on field rituals about the game are about respecting the game itself. Shaking Harbaugh's hand doesn't mean you have to like or respect Harbaugh, but it means you have to respect the game.
Unwritten rules to every sport. You don't put your feet under a player who is about to land after taking a jumpshot in basketball. You don't signal for time until the umpire has truly called you safe on a steal attempt. You don't run up the score after you have a wide margin of victory assured in football.
Would John Wooden do it? Would John Wooden think it was important to stop, shake hands and acknowledge each other after the game, even if you hate each other's guts? I think yes. The ethical coaching gold standard IMHO is John Wooden in all of sports.
How you feel should have no bearing on doing your job. Tomlin is miffed over a press statement? Who gives a flying ####,
do your job. Your job as coach is to be an example to the rest of your team for sportsmanship, even if you don't like it.