The Whirlwind, age 50, won two matches to raise hopes of a return to the venue where he was runner-up six times, but he found Gloucester's Milkins too tough an opponent.
Milkins led 6-3 after the first session with top breaks of 63, 76, 53, 71 and 50. And he added the four frames he needed in the evening session with a best run of 80.
"It is very disappointing after finding some form recently," admitted White. "I didn't play well at all out there - if I had got to 9-7 I might have had a chance. I will still be going up to Sheffield to support Ronnie O'Sullivan."
Peter Ebdon, the 2002 champion, came through a stiff test against Norway's Kurt Maflin, taking the last four frames to win 10-8.
Sam Baird made it six Crucible debutants this year with a tremendous 10-3 thrashing of Joe Perry. The 24-year-old from Cullompton in Devon, nicknamed The Blade, slashed past his experienced opponent with top runs of 78, 62, 77, 86, 53, 142 and 63.
Mark King denied China's Yu Delu a first trip to the Crucible with a nervy 10-9 success. Romford's King trailed 8-6 and 9-8 but nicked the 18th frame on the colours then won the decider with a break of 63.
Former Grand Prix and Shanghai Masters champion Dominic Dale eased to a 10-5 win over Alfie Burden with a top break of 131.
Dumbarton's Marcus Campbell ground out a 10-4 win over Liam Highfield despite a top break of just 52, while Betfair Shoot-Out winner Martin Gould came from 5-2 down to beat Rod Lawler 10-7.
