I think the game has specific criteria for which water spawns what...in order to safeguard against destroying your island's spawn rates. Like...I'm pretty sure that a "river" just means that it connects to the "mouth" at some point...and ponds are just anything dug in flat land with no connection to a river...and waterfalls are obviously waterfalls. Lol. But, yeah...I don't think the game would allow you to do anything that would ruin your actual ecosystem.
Kill it! The generator gave me a horseshoe-shaped pond that I thought I would really like but it was impractical, and long story short, you can make your own. ^^