I was cleaning a bathtub on the second floor and unfortunately left the water on with the sprayer on the floor. It ran for about 5 - 10 minutes before discovered. The water leaked through the tile floor and dripped through the ceiling of the first floor. After cleaning up the water, what is the next step? Do I have to worry about the stability of the 2nd floor floor/1st floor ceiling? What about mold inside the walls? Will it dry by itself?

Most likely you will only have water stain on th ceiling below. Not much else you can do but wait for it to dry. All depends on how much water. If it was a great deal you might want to make a small hole in the ceiling below and let the water out. That's about all you can do.

Harold is right, I might add that a fan in both the upstairs room and down stairs will speed up the drying. The faster you dry thingts out the better, mold will start growing in 48 hours if the ceiling is still damp.