While we withhold the right to moderate and maintain standards of discussion in our comments because they are a private space, does the President of the United States have a different standard to meet? The Knight First Amendment Institute at Columbia University believes so and sent Donald Trump a letter on behalf of Twitter users that have been blocked by his @realDonaldTrump account. The Wall Street Journal posted the text of the letter (PDF), which argues that, as President, operates as a "designated public forum" similar to a city council or school board meeting. As such, their argument goes, blocking users who have mocked or criticized the president violates their First Amendment rights.
The letter did not directly state that further legal action could follow, but certainly implied it's possible. Bloomberg reports that White House spokesman Sean Spicer confirmed in a press conference that Trump's tweets should be considered official statements. So far there is no comment on the letter, and the users remain blocked.