The pitcher was taken to a nearby hospital, then transferred to Banner Good Samaritan Medical Center in Phoenix. Kremchek said he expected Chapman to remain hospitalized for a couple of days, perhaps being released on Saturday.
The ball hit Chapman in one of the most protected areas of the skull, the doctor said.
"If you get hit in the side of the head, that could be disastrous," he said. "Where Aroldis got hit, you don't want to say he got hit in a good spot because he's undergoing surgery, but it could have been a lot worse, a lot more injuries, a lot more permanent. He's very lucky."
The 26-year-old Chapman has a fastball that regularly tops 100 mph. The two-time All-Star defected from Cuba in 2009 and made the Reds in 2010. He had 38 saves each of the past two seasons, with 122 strikeouts in 71 2/3 innings in 2012 and 112 strikeouts in 63 2/3 innings in 2013.
Price, a former pitcher, said pitchers are in a dangerous situation, "regardless of how hard you throw."
"It's hard to defend yourself from 53, 54 feet," the manager said. "And everyone finishes their pitches differently. Everyone is not in a perfect fielding position and even if you are there is no guarantee that you can protect yourself when a ball's hit that hard."
Major League Baseball approved protective caps for pitchers this winter following several terrifying scenes similar to this one in the past few years. The hats were available for testing during spring training on a voluntary basis.
"It's horrific (what happened to Chapman), and it just puts an exclamation point on the fact that pitchers need more protection," Bruce Foster, CEO of 4Licensing Corp, told ESPN.
Foster's company makes the isoBlox padded cap for pitchers in conjunction with official capmaker New Era. The cap met Major League Baseball's approval in January.
"I'm not sure our product would have helped, because of where he was hit, but surely the violence of these hits is tremendous," Foster said Thursday.
Foster says that based on some pitchers saying that the padded caps don't yet meet their standards in terms of appearance or comfort, his company is modifying them.
"I would hope that in the first two to three weeks of the upcoming season, all the players will have seen a cap, and be able to place custom orders for them," he said.
Chicago Cubs manager Rick Renteria, meanwhile, empathized with Chapman Thursday, related to his own surgery to fix a facial bone after being hit by a line drive. Renteria missed a season as a coach of the San Diego Padres after getting hit in the jaw by a liner during batting practice in 2011.
"I understand the feelings he's going through right now," Renteria said Thursday. "I hope he's doing well and recovers and is back on the hill as soon as he can."
Pena said Chapman particularly wanted to thank the Royals for their support and offer assurance to Perez that it is just something that happens in baseball and was not his fault.
Pena said he also felt some responsibility.
"I kind of blame myself a little bit because I could have called slider or I should have called changeup," Pena said. "That's your thought process. Everything goes through your mind and you're looking for answers. ... You kind of put yourself in that guilt feeling."
But Chapman, Pena said, told him "'You know, it's not your fault. I should have thrown slower.' I'm the one feeling very bad about it and he's the one that's cheering me up. He's the one in the hospital."
ESPN's William Weinbaum and ESPNChicago.com reporter Jesse Rogers contributed to this report.