No. It does have to be off the ground before he can toss it or pitch it. He can catch it and then get up and run.
Here is what an exception in rule 4-2-2 says:
ART. 2 . . . The ball becomes dead and the down is ended:
a. When a runner goes out of bounds, is held so his forward progress is
stopped or allows any part of his person other than hand or foot to touch
the ground.
EXCEPTIONS:
1. The ball remains live if, at the snap, a place-kick holder with his knee(s) on the
ground and with a teammate in kicking position catches or recovers the snap while
his knee(s) is on the ground and places the ball for a kick, or if he rises to advance,
hand, kick or pass; or
2. The ball remains live if, at the snap, a place-kick holder with his knee(s) on the
ground and with a teammate in kicking position rises and catches or recovers an
errant snap and immediately returns his knee(s) to the ground and places the ball for
a kick or again rises to advance, hand, kick or pass.
NOTE: The ball becomes dead if the place-kick holder muffs the snap or fumbles and
recovers after his knees have been off the ground, and he then touches the ground
with other than hand or foot while in possession of the ball.