WASHINGTON — Tensions over President Trump’s nominations turned into a parliamentary game of chicken on Monday, with Republican leaders vowing to keep the Senate in session, day and night, until lawmakers confirm four of his cabinet picks.

After days of grasping at procedural hurdles, Democrats held vigil against Betsy DeVos, Mr. Trump’s polarizing nominee for education secretary, and promised to spend the final 24 hours before her confirmation vote Tuesday reiterating their objections.

But as Democrats made clear they had no intention of yielding even a minute of their allotted floor time to debate Mr. Trump’s nominations — a final act of parliamentary disobedience for a minority party that lacks the votes to block a nominee on its own — Republicans stood their ground.