It passed in a 316-101 vote.
The New Democrat Coalition calls for fixing the tax code and raising infrastructure spending.
Nelson said he was "assured" the bill would not come up before the March 24 deal deadline.
Senators will vote a day sooner as a snowstorm heads for D.C.
"The Stand with Israel Act of 2015 sends a clear message," Paul said.
"Jon Stewart can’t write that stuff that good. He doesn’t have to," Rep. Gutiérrez said.
The Supreme Court is hearing oral arguements on a challenge to the health law.
Senators are expected to take a procedural vote Wednesday on overturning the veto.
The Senate will also take a procedural vote on overridding President Obama's veto of the Keystone XL pipeline.
Sen. Mitch McConnell (R-Ky.) said lawmakers could take up the legislation next week.