The Englishman won last week's Made In Denmark in stunning fashion with captain Bjorn watching on, but it was not enough to earn him one of four wild cards which instead went to Ian Poulter, Paul Casey, Henrik Stenson and Sergio Garcia.
Wallace confessed to feeling as though he'd done enough to warrant selection but Bjorn felt otherwise, instead packing his team with experience despite form and fitness concerns for some of his selections.
"I've had a lot of disappointments in my life, in my career playing golf, and I've always used it as fire to get better," said Wallace.
"This is probably one of the biggest (disappointments) you can have, so watch this space."
The 28-year-old, who began the Omega European Masters with a two-under-par round of 68, told Sky Sports that he was informed of the decision in a phone call from Bjorn on Tuesday.
"He called me up and he said how impressed he was, how well I've been playing - he was very complementary of my game and my mindset.
"He ultimately said it was not good news and I accept that. I respect his decision, I've looked at the people he's picked and he's gone with experience, which I believe they probably need as well.
"Obviously I'm disappointed, because that was my goal and my dream, and I thought I showed enough on the Sunday to get a pick."
Playing alongside Wallace was Thomas Pieters, who shot a three-under 67 to star in the marquee group which also included Matt Fitzpatrick, two worse on 69.
Asked if the trio discussed Bjorn, Pieters said: "We're like the rejects! We had a good day, all three of us. No Ryder Cup chat, all just normal, whatever we talk about on the course."
Fitzpatrick appeared frustrated throughout the first round of his title defence to sit on one-under, four adrift of compatriot Andy Sullivan who carded his lowest opening round since 2015.
Germany's Max Kieffer reached six-under in a round of 64, one ahead of Sullivan and Soren Kjeldsen, while Charl Schwartzel sat alongside Pieters, with Lee Westwood alongside Wallace following a closing bogey.
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