Shaw started for England in the 2-1 loss to Spain in Berlin.
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He said afterwards: "I am completely devastated. We really believed we could do it, but it wasn't meant to be.
"We had massive faith and the fans deserved it. I just want to thank them for their support and can only apologise to them."
The final was 29-year-old Shaw's first start since he was injured against Luton for United in the Premier League on February 18.