Apologizing is a sign of weakness. It is something losers do.
Trump is a winner.
The current President elect of the Philippines, who is often compared to Trump, did not apologize for a far more outrageous thing than anything Trump has said and it still didn't hurt him.
In the context of personal relationships, I would not agree with this. The problem with most apologies is that they are insincere, half-hearted, or apologies only for the effect and not for the action (e. g. "If I offended anyone, I'm sorry.").
But in the context of a rough-and-tumble political campaign, "No Apologies" is pretty much a guiding force. Our electorate does not appear to reward introspection.