You raise several interesting questions. The questions will undergo a modest redirection, but I hope that I will not lose the intent of the OP.
Does absolute pitch have a heritable component? Yes, according to this University of California Genetics of Absolute Pitch Study. Both European families and East Asian families had a “linkage peak” at 7q22.3. In an important sense, Yes, perfect pitch is something you are literally born with.
Is tone deafness something you are born with? Yes, according to http://www.newscientist.com/article/dn497-tone-deaf.html, which compared identical twins (who share 100% of their genes) and fraternal twins (who share 50% of their genes). The study suggested that “your genetic make-up largely determines your ability to perceive pitch. Spector's team concludes that this skill is roughly 80 per cent hereditary.”
This study would suggest that if you (or your identical twin) were tone deaf, then there would be very little chance that you could learn AP, no matter how hard you tried. Life has dealt you a weak hand; stay away from the musical arts. This would be similar to a comment made by a color-blind person. This sex-linked genetic trait will prevent him from a career in interior design. (While I am guessing that user unknown is male, the odds are in my favor.)
What if you’re not tone deaf but desperately want to impress that dazzling violinist? Can you learn how to fake AP as well as a Chinese concert pianist who started at the age of four? Maybe, according to Absolute memory for musical pitch: Evidence from the production of learned melodies by Daniel Levitin. You could work on your pitch memory, which is more common. Maybe you could conceal a hit tuning fork, hit it in the bathroom and hum the issuing note sub rosa before you come to the piano. Maybe you could have a mirror or confederate help you figure out what note is being played. Or, barring that, maybe you could memorize the answers for this AP test.