Roberts made his major tournament debut 12 minutes from time in Saturday's last 16 game against the Danes, but Wales were already two goals down and heading out.
Kasper Dolberg opened the scoring on 27 minutes from the edge of the area and doubled the Danish lead after the interval, smashing in after a Neco Williams mistake.
Joakim Maehle added a third before Roberts' fellow replacement Harry Wilson was red carded for a foul on Maehle. Martin Braithwaite added a fourth in stoppage time.
Escaping from a tough group thanks to a 1-1 draw with Switzerland and a 2-0 victory over Turkey, tipped by some pundits before the tournament to be dark horses, Wales qualified for the knockout rounds with a game to spare. A much-changed Italy side could only beat the Welsh 1-0 in the final group game, despite a 55th minute red card for Ethan Ampadu.
"That one hurt," Roberts posted on his Instagram account after the Denmark defeat.
"But we leave with our heads held high. We will be back, and stronger."
The 22-year-old added his gratitude for the support the Welsh have enjoyed throughout the Euro 2020 campaign.
Team-mate Chris Gunter, meanwhile, has used social media to lash out at a 'joke set-up' that saw Wales travel a total of 5,832 miles during the tournament, via Baku, Rome and Amsterdam.
"Every nation had fans wherever they went," Gunter wrote on social media.
"[But Wales did not] apart from the 350 who broke government rules and bank accounts to be there. You and us deserved more from this joke set-up of a tournament, but who said life was fair. Have a cry, but then smile that we were dining at the top table yet again."
Boss Page admitted the travel requirements, which did not impact other teams, was baffling.
"I don’t understand the format if I'm being completely honest," he said.
"We've travelled all around Europe and others haven’t. I'm not going to sit here and grumble about that. We'll take the defeat and we want to wish Denmark all the very best moving forward."