The amendment would have blocked $1.2 trillion in Medicaid cuts.
Sen. Chris Coons said Congress must pay for the fight against ISIS.
The Budget Committee leader said the chamber has taken 17 votes in more than six hours.
They unanimously backed sanctions if Obama can't verify that a deal is being followed.
The Senate is voting on amendments to the GOP budget.
Sen. Blunt said his amendment would "send a clear message" to the administration.
Sen. Scott said his amendment would raise awareness about a tax on premiums.
Senators backed year-round Pell grants for college students, but rejected a funding increase.
Senators rejected the measure in a 45-55 vote.
The paid sick days amendment was offered by Sen. Patty Murray (D-Wash.).