Afterward, Diamondbacks third baseman Mark Reynolds called Lee's actions "bush league."
With two outs in the bottom of the third inning, Lee threw an inside pitch to Snyder, then tossed one just over the batter's head.
Snyder walked toward the mound, and the dugouts and bullpens emptied. No punches were thrown.
Plate umpire Brian Knight tossed Lee but let Snyder stay in the game.
In the first inning, Lee collided with Snyder while trying to back up a play at home plate. But the left-hander denied throwing at Snyder intentionally and said he wasn't trying to retaliate.
"I was trying to go inside and a couple got away from me and the guy hitting got mad and came close to the mound and I got thrown out of the game in the process," Lee said. "That's it in a nutshell."
Lee didn't have much of an explanation for what happened in the first inning.
"I really don't know. I wasn't watching. All I know is I was just rolling on the ground, got knocked down," he said.
Acquired from Philadelphia in a big trade this offseason, Lee was tagged for four runs and six hits before leaving the game. He said it was the first time he'd been thrown out of a spring training game.
Lee said he went to a field behind the stadium and threw another 15 pitches to complete what was supposed to be a four-inning outing.
Snyder said he walked toward the mound "because he threw a ball at my head. You stand in there and have somebody throw a ball at your head."
Snyder, who missed much of last season with injuries, said it's no time for that kind of thing.
"It's spring training, man. Guys are going out there to get their work in, get their repetitions, get ready for the season," he said. "There's no time to start throwing balls at people and clearing the benches. There's no time for that in spring training."
Mariners manager Don Wakamatsu declined to comment about the dustup during Arizona's 13-5 victory.
Snyder said he thought he and Lee were OK with each other after their collision behind the plate.
"I didn't see a problem," Snyder said. "There's two guys going to where they need to be and we collided. He got me good, man. He Charley-horsed my leg. I can still feel it. ... My leg hurts every time I squat. Then he threw a ball at my head."
Reynolds was extremely critical of Lee's two pitches to his teammate.
"The first pitch he came up and in on him," Reynolds said. "We were hitting him around a little bit and the next pitch was just uncalled for in that situation. It's spring training. We've got a lot of young kids in the lineup just swinging the bats. If you're going to hit somebody, just stay below the waist. Don't headhunt. Don't do any of that. It's a bad reputation to have. It's bush league. Stuff like that doesn't need to happen anytime, especially in spring training."
Reynolds had no doubt Lee was throwing at Snyder.
"You're messing with guys' careers going up that high," Reynolds said. "I've faced Cliff Lee plenty of times to know he's got amazing control."