Are all cells organisms?
Your friend is incorrect. From Wikipedia:


  The word organism may broadly be defined as an assembly of molecules functioning as a more or less stable whole that exhibits the properties of life.


(Viruses are specifically excluded, as they depend on a host cell to fulfill all the functions of life.)

While cells extracted from a multicellular organism like a plant or a person may be able to be cultured in vitro for a time, they cannot survive independently - they require the intervention of humans (or very well-trained monkeys) to obtain nutrients and oxygen, and process/remove waste. A unicellular organism like a bacterium, for example, can handle these functions on its own - it can either synthesize or find a source for its own nutrients, and can reproduce on its own to create more organisms.

The key difference is being self-sustaining. An organism needs to be able to feed itself, take care of its waste, reproduce a full version of itself, respond to stimuli, etc. (see the "properties of life" link above). A single cell from a multicellular organism cannot do all that without assistance (kind of like a virus, actually), while a true unicellular organism can.