No, Romex (non-metallic sheathed cable) should not be exposed for under cabinet lighting or any other application. Romex is not designed for exposed installations and is not rated to be used in areas where it can be subject to physical damage or come into contact with moisture.
For under cabinet lighting installations, it is essential to use electrical wiring that is specifically designed for such applications. The appropriate wiring for under cabinet lighting is typically a type of armored cable called "BX" or "MC" cable (metal-clad cable). This type of cable has a protective metal sheath that provides better durability and protection against physical damage compared to Romex.
Alternatively, you can use flexible metallic conduit (FMC) or liquid-tight flexible conduit (LFMC) for exposed installations. These conduits provide a protective housing for the electrical wires and ensure compliance with electrical codes and safety standards.
If you are unsure about the electrical requirements for your under cabinet lighting project, it's best to consult with a licensed electrician who can assess your specific situation and ensure that the installation is done safely and in compliance with local building codes. Electrical work should always be performed by qualified professionals to ensure the safety of your home and its occupants.