The South Carolina delegation and black lawmakers led a moment of silence.
The bill now heads to the Senate.
The bill also cuts $8.25 billion from a preventive health fund.
Sen. Charles Schumer said Republicans "seem content to let deadlines blow right on by."
Sen. Tim Scott said that the families are "the best of who we are as Americans."
Senate Majority Leader Mitch McConnell (R-Ky.) said Tuesday that Kentucky should consider moving a statue of Confederate President Jefferson Davis currently housed in the state's Capitol.
"We curiously enough have a statue of Jefferson...
Many lawmakers plan to attend funerals for those killed in last week's shooting.
The Senate is expected to vote on going to conference with House lawmakers later this week.
"The United States is the only advanced country where this type of mass violence occurs."
"It's an important moment for the country," he said ahead of a procedural vote.