Chapman received a three-game ban, while Boone and Cash were suspended one game each. All received undisclosed fines.
"I don't think I should be sitting out tonight, nor Chappy, but I do respect the process," Boone said, adding that he didn't know if Chapman was appealing.
New York ended a six-game losing streak against the Rays with a 5-3 victory Tuesday.
Chapman threw a 101 mph fastball near the head of pinch hitter Michael Brosseau while closing out his first save of the season. Umpires met before issuing warnings to both benches amid a budding rivalry that has been dominated this season by Tampa Bay. The Rays are 7-2 against the Yankees and lead second-place New York by 3 games in the American League East.
Chapman struck out Brosseau to end it, and Brosseau began walking toward the visiting dugout before turning back to exchange words with someone on the Yankees.
The dugouts and benches emptied and players gathered near home plate, but they mostly kept their distance before dispersing to their respective clubhouses and umpires stepped in.
"Both teams are professional enough to know that we're playing for a lot," Boone said. "Even though there's been a few of these incidences where the blood gets boiling and tempers flare a little bit, both teams are smart enough to realize that it's about getting the job done on the field and performing -- and that's certainly our main focus and will continue to be. And I'm sure it is with them as well.
"Hopefully this is something that can all be behind us and it can be about two good teams going at it on the field."
Yankees starter Masahiro Tanaka hit Joey Wendle with a pitch in the first inning, which Cash said he believed was intentional, prompting a bit of a back-and-forth with Boone after the game.
"It's absolutely ridiculous," Cash said. "It was mishandled by the Yankees. Certainly the pitcher on the mound. It was mishandled by the umpires.
"Somebody has to be accountable. The last thing I'll say on it is I got a whole damn stable full of guys that throw 98 mph. Period."
While calling Cash's comments "reckless and inflammatory," Boone on Wednesday said he hopes "cooler heads will prevail and it can be about playing baseball."
Cash said the suspension for his comments was "fair, warranted."
"Three minutes prior to [those comments], we just had a guy thrown at," Cash said Wednesday. "So, fairly raw with what was being said. But always want to show to the guys how much we care about them as a staff, as an organization."
Teams have been warned not to leave the benches this season as part of the league's coronavirus protocols. Players on both sides seemed to be urging others to deescalate.
Chapman began the season on the COVID-19 injured list. He had been 0-for-1 in save chances this season.
The Associated Press contributed to this report.
Benches clear at end of Yankees-Rays
After Aroldis Chapman throws a pitch over Michael Brosseau's head, they stare each other down at the end of the game, tempers flare and the benches clear prior to Rays and Yankees exiting.