What certifications are required to sell a home kitchen appliance in the USA?
Certifications and legal requirements aren't the same thing.  There are no certifications that are actually required by law since your device does not contain a RF transmitter (those do need certification).

The only legal requirements are that the device not radiate radio waves above some limit, and that it not conduct signals back onto the power line above some amplitude limit.  In both cases, the maximum allowed amplitude depends on frequency.  The radiated emissions are regulated up to a few GHz, and the conducted emissions much lower.  The point of both these is to prevent your device from interfering with radio communication.

For a unintentional radiator, there is no requirement that it be tested and certified, only that it comply.  Getting it tested is a good way to know you aren't violating the limits, and things will go a lot better later on if some units are found to violate the limits if you can show due dilligence and proper testing.

In practise, you need UL or equivalent approval for any device aimed at the ordinary consumers.  If your device catches fire and burns down someone's house, or shocks them, things will go very badly in court if you can't show that "accepted design practises" were used.  Because not only will you get sued, but also the retailer, most retailers aren't going to touch your product without it having the relevant UL (or equivalent) certifications.