He went 2-for-3 with a single and a double in his final Gulf Coast game on Friday, a 5-4 loss to the GCL Dodgers.
To make room for Mientkiewicz, the Mets designated first baseman Brian Daubach for assignment.
Daubach, 33, hit .120 with a home run and three RBI in 15 games for New York.
Padilla, 28, represented Norfolk – along with starter Jason Scobie – at the Triple-A All-Star Game in Sacramento, Calif. this week. He was 3-2 with 11 saves and a 1.44 ERA.
Ring, 24, was 0-2 with a 5.06 ERA in 15 appearances. In 10.2 innings, he had walked 10 and struck out eight.
"We just feel like it's the right move," said Randolph, who added that Ring's demotion didn't have anything to do with Kazuhisa Ishii's relief status.
The Mets will make an accompanying roster move on Saturday to add Padilla to the club's 40-man roster.