Given how many vile things Trump says to others he has to be able to take it himself when people say vile things back to him. Of which most of the criticism aimed at him probably won't be put nicely and lots of it will be vile. Trump has openly criticised Obama even saying he was the founder of ISIS, when Trump says these things I have no doubt he will accept others have the right to say such crazy things at him.
President-elect Trump will absolutely be able to handle criticism, as he has routinely dealt with infinitely more toxic people in the upper echelons of businesses. Every single "your policies suck" or "you're racist", apart from being unsubstantiated, are NOTHING compared to what he's been working with for the last several decades. He would not even approach the Presidency without being able to control himself, so his election means that the American people have, for better or for worse, put their trust in him.
Trump has shown many times over that he is not okay with criticism. He simply cannot handle anyone talking ill of him or poking fun at him. He flies off on his Twitter tirades and makes sure everyone knows how he feels about anyone that dare cross him. Freedom of speech is not important to him.
All American have the right to criticize their president because of the first amendment of the US constitution which protects freedom of speech. However, Trump has made it clear during his campaign that he is thin skinned and takes all kinds of criticism very personally and reacts in inappropriate ways. He has also said that he wants to reform libel laws to limit what the press is able to say about him.
Trump has shown while campaigning that he has trouble with critics. He is quick to respond and that often results in childish comments from him. His lack of filter and access to social media often results in inappropriate exchanges with those of opposing views. He may never come to see it as a right of citizens, but hopefully he will learn to ignore it.