The team made the announcement just before the start of Monday night's game against Miami, which the Diamondbacks lost 3-2.
Kennedy got the most severe of the six suspensions stemming from the brawl between the Diamondbacks and Dodgers in Los Angeles last Tuesday. He was suspended after hitting Yasiem Puig and Zack Greinke with high inside pitches, the latter one triggering the bench-clearing melee.
Kennedy started Sunday's game in San Diego, leaving with no decision in a 4-1 Arizona loss. The right-hander, a 21-game winner two seasons ago, is 3-4 with a 5.21 ERA.
With off days included, Kennedy will miss one start and could have his second start pushed back a day to June 29, when he's eligible to return.
Kennedy said he and manager Kirk Gibson spoke "and we felt like this is the right time, the right thing for the team."
"Just to go through with it, and myself I felt the same thing," Kennedy said. "I kept on getting questions every single day and my teammates and everybody else. So I figured it was the best thing for the team so we could put it past us. "