With Medusa, you can attack an opponent, turn their units to stone, then Pillage as if they were not here.
You can also occupy their buildings. Since they are turned to stone until the end of the turn, you can use this to claim a title.
HOWEVER, what happens during next production phase ? Obviously there is no battle (not until the movement phase), but do you keep control of the buildings (and therefore their production ?
Similarly, could you try to convert a region where you Petrify an enemy, if you managed to kill the (back to normal) units on the NEXT turn ?
Here's the rules text for Medusa,
"Once per round, petrify opponent Legions when they enter
a Province occupied by at least one of your Units. Only you
roll Battle Dice.
Ex: If you only have a defending Fortress, you destroy one
Petrified Legion without having to roll the Battle Dice."
Sounds to me like Medusa is used for defense, not offense as your question poses.