Sen. Ted Cruz wasted little time in reacting to Monday's Supreme Court rejection of an Arizona law that demands would-be voters prove they are citizens, filing legislation hours later to allow the measure to return.

The high court voted 7-2 to throw out Arizona's voter-approved requirement that residents document their citizenship when using a voter registration form produced under the federal "motor voter" law.

The Texas Republican's measure, co-sponsored by Sen. David Vitter, R-Louisiana, says that "nothing in [the motor voter law] shall be construed to preempt any state law requiring evidence of citizenship in order to complete any requirement to register to vote in elections for federal office."

Cruz, a Tea Party favorite, filed the proposal as an amendment to a comprehensive immigration bill the Senate is debating this week.

"The right to vote is a fundamental building block of our nation's democratic process and it is crucial that we have the measures in place to uphold the integrity of our elections," Cruz said. "This amendment ensures that states can enforce the commonsense requirement that those registered to vote must actually be U.S. citizens."

Cruz says the measure doesn't require states to adopt certain voting requirements, "respecting the interests of federalism."

The senator didn't blame the Supreme Court for its ruling, saying the motor voter law passed by Congress in 1993 was flawed.

The law as written "encourages voter fraud and we must ensure that our elections are fair and accurately reflect the will of our citizens," he said.