In corresponding moves, the Yankees optioned infielder Dean Anna and right-hander Preston Claiborne to Scranton/Wilkes-Barre.
Greene, 25, is with the Yankees for the third time this season. He made his Major League debut in the seventh inning Friday's 14-5 victory over the Red Sox, recording one out while allowing three runs (none earned) on no hits and walking three, with a strikeout.
"We felt that needed a couple of pitchers," manager Joe Girardi said. "[David Phelps] is not available. Then we don't really have a true guy who can give you any distance down there.
"It's not the spot that you want to be in, but we felt that [Yangervis] Solarte, he can play shortstop a day or two if we need him to and he gives you a little bit more options being a switch-hitter."
The Yanks also transferred right-hander Ivan Nova, who will have Tommy John surgery on Tuesday, from the 15-day disabled list to the 60-day disabled list.
New York must play with a 24-man roster during Pineda's suspension; Pineda will be eligible to play on May 5 against the Angels in Anaheim.
Billings has pitched in four big league games with the Athletics and Rockies, all in 2011. The 28-year-old was 3-1 with a 2.74 ERA in four starts at Scranton/Wilkes-Barre, and he attended Spring Training with New York as a non-roster invitee.
Bryan Hoch is a reporter for MLB.com. Follow him on Twitter @bryanhoch and read his MLBlog, Bombers Beat. This story was not subject to the approval of Major League Baseball or its clubs.