The teams will play a traditional doubleheader on Saturday starting at 4:30 p.m.
Texas had a scheduled day off on Thursday, and the rainout enabled them to remain undefeated heading into the weekend.
"I wish I could get excited about that," manager Ron Washington said. "I'd much rather play, but since we can't, we'll go two tomorrow."
The players didn't seem to mind. Minutes after the postponement became official, Ian Kinsler and six other Rangers slid on the tarp.
Texas came to town after going 6-0 on a homestand against Boston and Seattle. If the Rangers can win the opener Saturday, they will match the best start in franchise history (7-0 in 1996).
A steady rainshower hit the area early in the day and never really let up. The scheduled 7:05 p.m. start was delayed for about an hour before the game was finally postponed.
"Everybody did everything possible to try to play," Orioles manager Buck Showalter said. "It just started moving a lot slower and settling in. You're going to have periods of 10 minutes when it's not raining very hard and then it comes back again. Plus, you don't want the fans to sit in that stuff, either."
Shortly before the game, Baltimore second baseman Brian Roberts experienced persistent stomach pain and was taken to the hospital for tests. The Orioles hoped it was just a virus that's been going around.
Baltimore is 5-1. Hours before the scheduled start, Showalter expressed hope that the game would be played Friday because he feared what a doubleheader would do to his pitching staff.
"You always have to think about down the road, the next day and the next day," he said. "If we play 18 innings (Saturday), that's a lot of baseball to pitch. And one of the things about being home, you know you're going to pitch nine innings."
The day off will give shortstop J.J. Hardy some time to bounce back from a sore left rib cage. Hardy was a late scratch Thursday against Detroit and omitted from the starting lineup on Friday.
"I've never had anything like it," Hardy said. "But being able to throw, being able to run around not feel pain, I feel like it can't be that big a deal. So, I'm hoping to be in this series at some point."
Showalter will start rookie left-hander Zach Britton in the opener and Jake Arrieta in the second game. Britton was scheduled to start on Friday night.