Which US president held the first White House press conference?

Woodrow Wilson!

The first White House Press Conference was held at 12:45 on 15 March 1913. Wilson had been president for 11 days and set up the news conferences as a way to let the press know better the president's thoughts on issues of the day. The very first conference was a rather awkward affair with 100 reporters and a lot of silence. They got a little better as Wilson got used to the format and the reporters got used to their new roles. Today's press conferences can be a lively affair which are quite different than the first ones over 100 years ago.

https://www.bbc.com/news/magazine-21761429