You cannot grow a new plant from the broken off bit, maybe put it in water as a cut flower and enjoy the blooms for a little while.
Find some complete Phalaenopsis culture notes on the net that are written for the climate type where you live in the USA.

I've done that to 2 of my phals this fall. As Aurthur stated, the broken tip can't be propagated by us mere mortals. If you're lucky, your phal may send out a side shoot below where it was snapped (probably below the existing bud) or maybe even develop a entirely new spike from the base of the plant.

The breaking of a flower spike shouldn't harm the plant in any way, so just keep giving it the care you would normally(I'm assuming you're in the USA) : indirect light (bright east window or shaded south/west window), 60 degrees minimum (temps us humans like), water weekly with weak fertilizer (1/4 to 1/2 recommended strength) and let excess water drain away.

Good luck,

Bob