The prolonged shutdown also puts millions of Americans at risk of missing their November food stamp benefits.
Mr. Trump places the blame for the shutdown at the feet of Democrats.
"The Republicans are voting almost unanimously to end it, and the Democrats keep voting against ending it," Mr. Trump said.
Mr. Trump has talked about fixing the health care system for years, including during his first term in office.
Mr. Trump helped end previous government shutdowns by bringing members of Congress to the White House for discussions.
"I'm not going to do it by being extorted by the Democrats who have lost their way," Mr. Trump said.
Mr. Trump said he believes that Democrats will eventually capitulate and vote to end the shutdown. »