Two New York police detectives were shot in the line of duty in December.
The GOP dissenters included hard-line conservatives and centrists.
The measure lays the groundwork for restricting funds to the UN if it discriminates against Israel.
The amendment would have extended sequester relief for two years.
Senators blocked a Democratic attempt to overcome a procedural hurdle.
An amendment to boost defense spending was adopted by a vote of 219-208.
Bernie Sanders's push to recognize man-made climate change was voted down.
Only Ted Cruz didn't vote for a measure from Debbie Stabenow.
The Progressive Caucus, Black Caucus and House Democrats each offered alternative fiscal visions.
The RSC budget would cut $7.1 trillion in spending over the next decade.