The two barreled into each other while going after Miguel Cabrera's fly out to deep left-center field. Coats managed to hold on to the ball for the out, despite both players' tumbling backward after the collision.
Coats threw the ball back and remained down in the outfield for several minutes while White Sox training staff checked on him. He then left the game, bleeding from the mouth and appearing woozy.
He was tested for a concussion and needed five stitches.
"I was more just dazed a little bit. I'm fine," he said. "I did some tests back there and there was nothing wrong. Everything was coherent. No problems. Just a little painful. Had a pretty good cut below my teeth."
Coats, 26, was hit by a pitch from Tigers starter Mike Pelfrey in his second at-bat of the game.
"It's not what I wanted, but I'm still happy I got to go out there today," he said. "It kind of stinks I had to come out of the game, but I feel fine."
Asked if he saw Shuck coming, Coats added: "The gap was kind of open and it was kind of no-man's land, and I didn't hear him. It got pretty loud. I started calling it. We just both went to the ball. It was just one of those ones that was kind of gonna fall in between. Just happy I caught it and got it back in, so that's all that matters."
The former 29th-round draft pick was called up from Triple-A Charlotte and inserted into the White Sox's lineup on Saturday, with outfielder Melky Cabrera on the emergency family leave list.