Presidents aren’t supposed to get into twitter battles. They have much more important things to spend their time doing, like managing a national pandemic response, working to ease racial tensions, creating and implementing strong foreign and domestic policies, reaching across the aisle for common ground legislation, uniting a politically fractured country, presenting a strong yet dignified image of the world’s economic leader, making decisions that cost the lives of American soldiers and soothing the pain of the ones they leave behind, keeping the nation's secrets...all the while getting enough sleep, food, drink, exercise, and downtime to stay on top of it all. Also he has to raise money for his re-election campaign and try to increase his voter base.
Twitter wars aren’t just a waste of time, they’re actively counterproductive to several of the goals above, and engaging in them shows a lack of priority management that is immoral. It also shows that he is uninterested in adjusting himself to the demands of his job, and would rather make it more difficult to accomplish his goals than acquire some grace and humility.
This is all tied to morality, and is only a fraction of the case against the existence of his moral compass.