I think you did the right thing by quickly interfering and separating the dogs before it escalated beyond a bit of shouting. I don't think you need to continue ignoring the dog when you get back home because by then he won't still be associating the punishment with the incident. Dogs don't link cause-and-effect to such an extent, so he may not even know why you are ignoring him by that stage which is a) confusing for the dog and b) not teaching him anything.
The short answer to your question: Treat the dog with calm authority after an incident like this, which seems to be what you have done.
I do have a further suggestion though, to prevent future issues and reinforce good behavior:
My suggestion would be to start as you did by separating the dogs. But from this point, rather than walk away from the park, walk them slowly away from each other until they calm down.
Once they have calmed, ask the other owner if you can walk a bit with them. Explain that you want their interaction to end on a calm positive note so if they meet again they won't instantly have a negative association and reaction to each other. Hopefully the other owner will be co-operative.
Walk them together, parallel to each other, and both on lead, with no balls or toys or treats to argue over. Leave as much space as you both feel comfortable leaving, but aim to make sure the dogs are close enough to know this is a joint activity. Walk them together for a while, and if you feel comfortable let them get closer to each other.
They don't have to be best buddies by the end of this walk, it could be less than 2 minutes and you can then go on your way. Or you can continue as long as you like to eventually let them off the lead to play together again if you judge that it's the right time.
Speak in low, soft tones, keep everything calm. Praise them both gently but only if they both remain nice and calm.
In this way, they will have forgotten the incident and next time they meet it will be a positive association.
Plus, you won't have to continue the punishment of ignoring or being stern for longer than it takes to walk home.
EDIT: I believe the above will only work if the scuffle did not turn into a bloodbath. If it had been a true fight with real biting (sometimes called red-zone) your dog could be high on adrenaline for days. No amount of correction or positive reinforcement will sink in, and it's best to just take him home and avoid over-stimulating his senses for a few hours / days.