I would use "apologize". "To apologize" is the infinitive, which doesn't quite fit in this case; "apologizing" is the participle/gerund, which definitely is wrong here. I would just use the present tense verb "apologize". As for other mistakes, the sentence is technically correct, but it's also very wordy; it doesn't sound quite like a native speaker would put it. I would say: "After such rudeness, the manager should apologize publicly." There's no need for the phrase "do what he ought to do"; if the manager "should" do something, then we already know that he "ought to" do it. I hope this was helpful! Good luck as you continue to learn English!

The simple and most common phrase would be : "..., the manager should apologize publicly." You could keep your structure and replace "to" with "and". Then it would be grammatical, but the style would still be very poor.

Keep it simple, as Michael suggested. You can almost never go wrong with simplicity in cmmunication.