I have grown a new root, not a tap root, but a new root on a seed I destroyed the tap root on (WW auto). I new I killed it, but I had to try to save it, I smashed the tap root picking it up, a bit of TLC and she made a nice harvest.
As far as growing a new tap root, no it wont happen, but we make clones that never have tap roots. I dont like it when I do damage to the roots, but they can survive.