I need a new battery and I can get a 500w 36V 14Ah naked battery for $129.00 but the bike has a 36v 250 watt motor. Will I do damage to or diminish the life span of the motor increasing the wattage from 250w to 500w?
The watt output is a measure of current. As long as there is sufficient rating to supply your controller you'll be fine. But you need to verify those numbers.
Some of those packs have a pretty low max-current output rating. Since that one is ebike-rated it's probly been measured and has a sufficient rating.