They two expressions can be synonymous but they are different expressions. "Stoop so low" is generally used in the negative, I believe, and is usually a response to some accusation of a slightly immoral act. For example, "I would never stoop so low as to charge my italki students for a text message between classes." The physical act of "stooping" (i.e. bending down at the knees" is not what is happening here. This is a figurative use of stooping. (By the way, stooping is something that birds do, more than humans, in my mind, anyway.) "Fall so low" can be figurative, but can also be very literal. "The water level of the reservoir fell so low that irrigation was no longer possible" for example. The (free) Merriam online dictionary spells out all these uses and more. I hope that is helpful.

Hi Oscar, I've never heard 'to fall so low' used interchangeably with 'to stoop so low', and it's not a collocation I would recognise in British English, I'd always use stoop rather than fall in that phrase. Regards, Paul