The measure requires recall elections to involve a party-specific primary followed by a general election.
The move is an apparent reaction to the ouster of Republican Russell Pearce from the Senate after losing a recall election last November.
Recall elections currently consist of one general election open to candidates and voters of all parties.
The measure failed by a vote of 3-6 Thursday, with three Republicans joining Democrats in opposition.
Among other objections, some lawmakers voting against the bill felt it was unlikely voters would approve it.
A similar bill is moving through the Senate.