Indians general manager Chris Antonetti said Saturday that Gomez won the competition to be the fifth starter.
Gomez has a 1.97 ERA in six spring outings. The right-hander left a start Friday against Cincinnati after slightly straining his right hip on a fielding play, but Antonetti said the injury is not serious.
Slowey was acquired from Colorado this offseason. He was optioned to Triple-A Columbus along with reliever Frank Herrmann.
Slowey went 0-3 with a 6.75 ERA in exhibition play. Herrmann, who split last season between Triple-A and the Indians, was 1-1 with 5.40 ERA.