President Trump on Tuesday urged Britain to forge ahead with plans to quit the European Union, reiterating his support for Brexit at a time of acute political crisis, as the prospect of a Britain overly dependent on the United States divided politicians and further inflamed thousands of anti-Trump protesters on the streets.

Mr. Trump dangled the prospect of a “phenomenal” trade deal with the United States if Britain breaks from Europe. But he made clear that in negotiations for such a deal, “everything will be on the table,” including opening Britain’s public health system to American competitors.

Taking credit for having predicted the outcome of Britain’s referendum in 2016, Mr. Trump said he believed Britain could now leave the European Union, even if it fails to make a deal with Brussels by the Oct. 31 deadline.