The vote was prompted by a several-hour debate in which members of the House of Representatives were allowed to make statements - many of them very emotional.
- We must preserve the United States as a republic.
We can not turn back the clock and erase the last four years.
But we can be inspired by the ideals of great presidents.
Like Washington, Jefferson, Lincoln and Roosevelt, said Democrat Steny Hoyer of Maryland.
Decided in the last stone
But it was not exclusively Democrats who then voted for the Supreme Court - ten Republicans voted to proceed with the procedure.
The process will now be taken up in the Senate, where support needs to be greater to overthrow Donald Trump.
In the House of Representatives, a simple majority is enough - in the Senate, a two-thirds majority is required.
How the Republicans in the Senate will vote remains to be seen - but earlier on Wednesday, the Republican Senate's Mitch McConnell opened to overthrow Trump when it emerged that he wrote in a letter to colleagues that he has not decided how he will vote.