I see a genuine, warranted apology as a sign of strength.
I didn't remember it exactly as it is in the movie so I looked it up. In "She Wore a Yellow Ribbon" John Wayne states to a subordinate in the calvary:
"Never apologize, Mister, it's a sign of weakness"
I think it's a good philosophy in politics. Your political enemies will use it against you. If you've been a little heavy handed with an opponent there are ways to be conciliatory without using a formal apology. The adults in the room will know what you are doing. Your enemies will have a harder time brow beating you with your own words.