Amaro said in a statement the recent tornado activity near Oswalt's home resulted in "several tragic deaths and significant devastation to the area."
"He has chosen to take time to make sure there wasn't significant damage to his home, but more importantly, to make certain that his wife and children are OK," Amaro said.
"It is almost a year to the day that his parents' home was destroyed by a tornado which has heightened his concern about the current situation."
Amaro said the Phillies hope Oswalt will be able to rejoin the team for his next start but the team will "take that day by day."
Oswalt threw three ineffective innings in a loss to the Arizona Diamondbacks Tuesday night and was long gone from the Philadelphia clubhouse when the game ended.
After the game, manager Charlie Manuel said Oswalt said nothing about being injured. Still, Manuel said, his ace right-hander "wasn't right."
A source who was in attendance Tuesday at Chase Field told ESPN.com's Jayson Stark that Oswalt was observed speaking on the phone in the dugout for a long time before the game.
Players rarely bring phones onto the field at any point in the day and many teams prohibit the practice.
Oswalt (3-1) allowed five runs on six hits with no strikeouts and one walk. He threw 57 pitches, 36 of them strikes.
"His stuff wasn't good," Manuel said. "His stuff wasn't crisp and his velocity was down. I figured there was no sense in leaving him out there."
Manuel said Oswalt didn't say anything about an injury. Still, the manager said he was a "little concerned."
The right-hander had given up five runs in 24 innings in his first four starts this season and had not been the losing pitcher since a July 30, 2010, outing at Washington.
Information from ESPN.com's Jayson Stark and The Associated Press was used in this report.