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From the Core Rules, page 220:
Core Rules wrote:
Adding Spells to a Sorcerer's or Bard's Repertoire: A sorcerer or bard gains spells each time she attains a new level in her class and never gains spells any other way. When your sorcerer or bard gains a new level, consult Table: Bard Spells Known or Table: Sorcerer Spells Known to learn how many spells from the appropriate spell list she now knows. With permission from the GM, sorcerers and bards can also select the spells they gain from new and unusual spells that they come across while adventuring.
Again, that's a quote from the Core Rules, page 220. (Emphasis mine.)
With GM permission (as allowed by the RAW without ever invoking Rule 0), your bard absolutely can learn gravity bow.