Can one become a scheduled airline pilot with just one functional eye?
Under current FAA medical regulations a pilot with one functional eye (called monocular vision) can get any class of medical, including the class 1 required for airline pilots:


  An individual with one eye, or effective visual acuity equivalent to
  monocular, may be considered for medical certification, any class,
  through the special issuance section of part 67 (14 CFR 67.401)


As the same page says:


  it has been repeatedly demonstrated that binocular vision is not a
  prerequisite for flying


Incidentally, one of aviation's most famous pioneers, Wiley Post, had only one eye. As for today, it's legal and possible to get any class of FAA medical with only one eye and therefore in principle you could fly for any US airline. Whether or not an airline would hire you is a different question, but if you're legal and qualified then why not? Blindness in one eye isn't even considered a disability by the US government if the other eye is fine.

It's also possible that if an airline declined to hire you because of your monocular vision even if you hold an ATP and class 1 medical, you might have a case for suing them for discrimination. But that's speculation, and you would need to ask another site about that. In any case, there may be more pilots out there than you think with physical limitations (for want of a better word), including ones with prosthetic limbs.