Oswalt told Washington after throwing 30 pitches in the seventh and eighth innings on Sunday that he couldn't continue. The Rangers lost 7-6 in 10 innings at Kansas City.
"Since I've been here (if) a player tells me he can't go, that's it," Washington said before Monday night's game against the Boston Red Sox. "It ends right there and, for me, it ended right there."
Oswalt had a 6.49 ERA in six starts and went to the bullpen when Texas acquired Ryan Dempster from the Chicago Cubs last Tuesday. Oswalt had thrown 150 pitches in seven days when he told Washington he couldn't pitch.
Rangers CEO Nolan Ryan also weighed in on the issue Monday, telling ESPN Radio 103.3 that he has not spoken to Oswalt about the situation.
"Those kind of things need to be left to management of the team, that being Ron and his coaching staff," Ryan told the radio station. "I don't know if there's an issue there or not, or if Roy was just to the point where he didn't feel like he was going to be effective or what. I try not to get involved in those things. ... That needs to be done within the ballclub."
Ryan would not commit to a specific role for Oswalt, stating that he can foresee the three-time All-Star pitching either as a starter or reliever.
"Roy's been very effective out of the bullpen," Ryan told the radio station. "I'm not going to go on record and say I think he's one of our bullpen or relievers, but under this circumstance he's been very effective there.
"If (Washington) and (pitching coach) Mike (Maddux) feel that's where we need the most help, I can see him staying in that role. If they feel they need him in the starting rotation, I would say that's a possibility."
Information from The Associated Press was used in this report.