Yes a bad battery can ruin a starter.
If the battery is bad and not putting out enough amperage to turn the starter over properly then the starter can suffer damage.
The starter motor needs enough amperage to properly spin over and start the engine.
If your battery is always low and the starter turns over slowly and you continue cranking the engine over with the bad battery the starter motor can heat up.
When the starter motor heats up the winding's in the starter motor get bad and then the starter motor can seize up and fail to turn over at all.
So if your battery is bad you need to replace it with a new one before it causes more expensive damage.
If the car battery is 5 years old or older it's time to replace it because car batteries do eventually go bad and I always replace a car battery or my truck battery after 5 years as preventative maintenance.
I would rather spend money now and have a new reliable battery the get somewhere and my car or truck fail to start.