It is most likely an Americanization of the English “to eat humble pie.” The English phrase is something of a pun—“umbles” were the intestines, offal and other less valued meats of a deer. Pies made of this were known to be served to those of lesser class who did not eat at the king’s/lord’s/governor’s table.
It may also be the American version of "umble," since the Oxford English Dictionary defines crow (sb3) as meaning 'intestine or mesentery of an animal' and cites usages from the 1600s into the 1800s (e.g., Farley, Lond Art of Cookery: "the harslet, which consists of the liver, crow, kidneys, and skirts."
Rudyard Kipling uses this expression, which predates his birth, in his short story “The Strange Ride of Morrowbie Jukes” (1885). Morrowbie Jukes falls into a place from which he cannot escape. Another man trapped there catches wild crows and eats them, but Morrowbie in his pride declares, “I shall never eat crow!” After days of nothing to eat, his hunger and desperation finally force him to do what he swore he would never do: literally eat crow.
In Rick Cook's Wizardry series, his main character is served crow by his wife after admitting that he had been neglecting her. His response is "Now I understand that phrase".
In Philip Roth's 2007 novel "Exit Ghost" the character Kliman talks to Nathan Zuckerman over the phone about the reelection of G.W. Bush: "It's a dark day, Mr. Zuckerman. I've been eating crow all morning...."
This idiom is used in the popular quote, "It's easier to eat crow while it's still warm." (Attributed to motivational speaker Dan Heist)