It may sound strange, but gravity stops the object from falling to earth, because by the time the force of gravity has pulled the object in a certain direction, the Earth has moved away from that point, pulling it in an endless loop around and around.
Put simply, the object is falling toward Earth, which is always changing location, and the satellite keeps changing its direction to find Earth. Sadly, by the time the satellite gets there, Earth is gone, and is pulling somewhere else.
(BTW, centripetal force isn't actually real, it doesn't exist. So be wary of those who say its the centripetal force. It only exists because we imagine it to exist.)