Surely there is a spectrum here from completely tone deaf - I have met a tone deaf student and so apparently have others - to possessor of perfect pitch, whether aural or vocal. Both are extremes of the "normal". The question is whether anyone can get better. I suspect there are some people who cannot - my example of colourblindness applies, some people cannot distinguish colours at all and see the world in tones on black and white, and these cannot be taught to see colours. You try distinguishing between dark blue and dark red when you are looking at a black and white printout.

A lot of people though can be taught to improve. Listening exercises will train the ear. Over many years I have taught myself what my throat feels like when I sing a high E or a low E, and can pitch those notes close enough for unaccompanied singing. I have also learned through many years of practice to sing intervals like 4ths and 5ths. But IMHO the answer to the question "Can the tone deaf learn to sing" is probably No - depending on your definition of singing!