Is it possible to determine whether a purchased phone has a new battery? I recently purchased a Samsung phone that was listed as "factory refurbished".  Something I found on Samsung's website says that factory refurbished phones will have a new battery installed.  However, this particular phone was being sold by a 3rd party, so I would like to verify whether the battery is actually new when the phone arrives so I can take advantage of the return window if necessary.

Is it possible to determine whether the battery is new, or to gain any information that would provide evidence one way or another, using either built-in features or a third party program?
If the battery is user replaceable, you can check the production number on that and try to decode battery date. I don't know if the manufacturer numbering scheme is standardised or if the app can be trusted to decipher, so I would suggest the alternative below. 
Edit  Accubattery Pro gives you an indication of estimated Vs actual capacity . I have no idea how accurate it is but surely can be indicative
A reliable indicator is the battery capacity, since capacity reduces with age. A battery, even if not used loses it's capacity to hold charge, significantly, over a period of time. You can refer to question for details
How to determine maximum capacity of a battery in mAh? - received fake battery
If device is rooted, you can check if information here helps  /sys/class/power_supply/battery/