WASHINGTON—With stitches prominent above his left eye after recent brain surgery, Sen. John McCain (R., Ariz.) returned to the Senate Tuesday determined to challenge his own party by calling for a return to the decorum and courage that have often defined the chamber.

“I hope we can again rely on humility, on our need to cooperate, on our dependence on each other to learn how to trust each other again and by so doing better serve the people who elected us,” Mr. McCain said on Tuesday to colleagues who listened in near silence....