Nope. He verifiably does not. The only time Trumps sued the press was when Melania sued Daily Mail for libel. Daily Mail lost and paid her $2.9M. https://www.snopes.com/news/2017/04/12/daily-mail-melania-tr...
If Trump sued for libel every time (which, IMO, he should, to provide some negative feedback to the system that's otherwise unstable), he'd put half the press in the country out of business by now.