WASHINGTON—House Democrats came up short Tuesday in an effort to override President Trump’s veto of a measure aimed at rolling back his declaration of a national emergency.

The 248-181 vote, which didn’t clear the two-thirds threshold, was the coda of a long struggle over funding a wall along the border with Mexico. Both the Democratic-led House and GOP-controlled Senate had previously passed a resolution rejecting Mr. Trump’s declaration of an emergency at the southern border, prompting him to issue his first veto earlier…