Davis finished one shot ahead of American Chad Campbell as heavy rain forced the qualifier to be reduced from 36 holes to 18.
No longer exempt after falling out of the world's top 50, Garcia was on level par when he quit after four holes.
Runner-up to Padraig Harrington at Carnoustie four years ago after bogeying the final hole and then losing a play-off, the 31-year-old Spaniard could still make it to Sandwich in July through upcoming events. But now 73rd in the world - he was second behind Tiger Woods just over two years ago - he has also entered a qualifying event in Memphis on 6 June for the US Open the following week.
Joining Davis and Campbell in July will be Australia's Nathan Green who shot a 66, a score matched by Americans Davis Love III, Spencer Levin, Chris Tidland and Bob Estes.
The final spot on offer went to a play-off after Americans Justin Hicks and Jerry Kelly both shot 67s. Kelly finally took the honours to fill the eighth qualifying position.