Democrats in the Arizona Senate forced a vote on legislation requiring background checks for any firearm purchase but it failed to pass.
Tuesday's procedural move came a week after a man armed with two AR-15-type rifles killed 19 children and two teachers at an elementary school in Uvalde, Texas.
The Democrats' effort would have closed the so-called gun show loophole that allows someone to buy a gun from a private party without the federal background check licensed gun dealers are required to perform.
Democrats pleaded for support for the measure, saying it would save lives.
Republicans said it would not keep a criminal from getting a gun and would prevent law-abiding citizens from getting a firearm.