Richards, making his second major league start, recoiled after throwing a 1-and-2 pitch to Nelson Cruz, his 19th pitch of the game.
Bobby Cassevah relieved Richards, striking out Cruz on his first pitch to end the first inning.
The injury to Richards puts the Angels in a bit of a jam as they try to stay in the AL West race. They entered Monday's opener of a four-game series four games behind the first-place Rangers. But that gap grew to five games as the Rangers beat the Angels 8-4 on Monday night.
Richards, who had gone 12-2 at Double-A Arkansas this season, but then lost his major league debut last week at Yankee Stadium, was the second rookie the Angels plugged into their rotation in place of a struggling veteran. Richards was replacing Joel Pineiro (5-6, 5.48 ERA), who was moved to the bullpen two weeks ago. Tyler Chatwood took the place of Scott Kazmir, who was unconditionally released in June.
"Our depth on the starting pitching side has been stretched," Angels manager Mike Scioscia said before the game.
Mark Saxon covers the Angels for ESPNLosAngeles.com. Information from The Associated Press was used in this report.