I had a satellite directly over one of my cities. In a nearby city I created short range missiles, launched them and then sent the missile to the square occupied the satellite. There was a battle and the missile fails to bring down the satellite. Is this how I am supposed to attack satellites?  Wrong weapon? Did I hit my own city?

I've always(?) used an armed satellite against another satellite... and I know that I've taken out an enemy satellite over one of *their* cities.

Check the battle report and see what it says about who attacked what and what the results were.  If you click on the battle result it will show you all the random "rounds" that took place, the probability of the combatants, who won which round, and what the final result was.

FYI:  When using a satellite to attack an opponent unit you use F - Fire to "bomb" the other satellite, you don't ram it.