The much-hyped Mexican-American, 23, knocked Dominican Fortuna down in the fourth and fifth rounds before stopping him early in the sixth in Los Angeles.
Garcia’s record reads 23-0, while Davis has won all 27 of his fights.
“If he wants it, let’s get it,” Garcia told DAZN. “That’s going to give me the respect I deserve.”
Inactive for 15 months, Garcia now has two wins in three months after beating Emmanuel Tagoe in April.
He challenged Davis to a December bout after watching the American beat Rolando Romero in June.
Davis tweeted after Saturday’s fight: “See y’all the end of the year.”
Garcia knocked Fortuna down with a left hook to the body, similar to the shot that stopped Britain’s Luke Campbell last year.
He dropped the 33-year-old again in the next round and ended the fight at the Crpyto.com Arena with a left hook to the head.
“I was crisp. I was sharp,” Garcia said. “I know how good a fighter I am. I just had to put it all together and you saw it come together today.”
Davis holds the regular WBA lightweight belt and American Devon Haney the more prestigious Super lightweight belt.