A steady downpour and low temperatures were predicted for much of the night.
No makeup date was immediately announced, although it was determined that the game will not be part of a doubleheader on Wednesday. Baltimore won the opener of what was to be a three-game set 7-1 Monday, and the series will conclude Wednesday afternoon.
The Rays were looking to break out of an offensive skid in which they scored only 14 runs in eight games.
Tampa Bay announced Tuesday that left-hander Matt Moore will be miss the rest of the season after opting to undergo elbow-ligament replacement surgery. It was a significant blow for a team that already has starters Jeremy Hellickson (elbow) and Alex Cobb (oblique) on the 15-day disabled list.
Moore had never missed time before, and Maddon knew it would be tough on the left-hander.
''Emotionally it's got to be very difficult for him right now,'' the manager said. ''It's going to be hard to ask him to be patient as he goes through the process, but we have great medical people here.''
Odorizzi will pitch Wednesday, Maddon said.
The Orioles, meanwhile, were preparing to play without center fielder Adam Jones, who was sent home with flu-like symptoms. Reliever Evan Meek was sent home with the same bug.
Last week, reliever Brian Matusz went to the hospital after falling ill, and third base coach Bobby Dickerson left the team for the same reason.
''It's not just in our locker room,'' manager Buck Showalter said. ''Unfortunately, medically, a lot of it is unavoidable other than don't touch or communicate with anybody all day. That's pretty hard to do in our sport. I'd like to say I haven't seen it before, but I have. I just hope it runs its course.''