The batteries won't damage the ESC by themselves, but the prop/motor/battery combination could. Ultimately, it is the amount of current that the motor tries to draw that is the issue. If the prop/motor combination try to draw more than 60 amps, and the batteries are capable of supplying it, then the ESC will fry. Of course, if the motor itself is rated at less than 60 amps, then it will fry first.
There really is not enough information to tell if the combination will work or not. What motor are you using, with what prop? What is the KV rating and current limit of the motor. Also, are you using 3S or 4S battery packs?