Several immigration/amnesty bills died in the House before Cruz was even elected to the Senate. The Rubio/Schumer bill died for the same reason the others died: the House is more responsive to what the people desire.
Cruz was not responsible for the most recent amnesty bill dying in the House. Rush and Levin need to review the history of amnesty bills since around 2006.
I don't know if Cruz is responsible for stopping it, but Cruz did clearly state in 2014 that the House Plan was a legalization amnesty, and that he was against it.