Their successes followed the bronze medal won by the Scottish women's team on Saturday.
Ten nations competed in team, individual all-around and apparatus disciplines at the two-day event, the first major international competition to be held at the newly opened Emirates Arena.
The top-eight gymnasts in each apparatus went head-to-head in yesterday's finals, with Scotland represented in eight of the 10. Regan finished second in the women's vault final, with Davie the runner-up in the men's high bar.
"It feels amazing to get silver on vault," said Giffnock-born Regan, who competes for City of Glasgow and finished eighth in the beam and A-bars finals. "I didn't expect a medal today so it's been a bonus and a nice surprise."
Davie, from Coatbridge, revealed he had been intent on making the most of a second chance after another competitor was forced to drop out.
"I'm so chuffed considering that yesterday my high bar routine didn't go well and I failed to make the final," said the 19-year-old.
"That was disappointing but another competitor had to withdraw due to illness. I was determined to use that opportunity.
"The home crowd has been amazing. They have been so supportive. That lifts you and makes you want to do well.
"It's great to get this experience and it has given me a little taste too of what things will be like at the 2014 Commonwealth Games. It's definitely motivated me."
Finland's Tomi Tuuha took home gold from the men's floor, bronze in the rings and silver on vault. Helge Vammen (Denmark), crowned men's individual all-around champion on Saturday, took gold on pommels.
In the women's apparatus finals Emma Larsson (Sweden) added gold on vault and the A-bars to her bronze in the individual all-around.
Davie finished sixth on floor and seventh on parallel bars with team-mate Douglas Ross sixth on vault. Fellow Scot Jordan Archibald came fourth in the women's vault and Cara Kennedy fifth on floor.
On Saturday, the Scottish women took team bronze behind Finland and Norway. The men's team finished in sixth place.
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