Martinez, a 22-year-old right-hander who is vying for a spot in the team's rotation, found himself in the middle of a Twitter firestorm this week after pornographic photos and links on his "favorites" page were discovered throughout his account.
While maintaining that his account was hacked, Martinez acknowledged the situation and worked with team officials to clean up his account Monday. But by the time he did, it was too late; the content went viral after being posted on Deadspin.com and spread by users.
The Cardinals expressed regret about the situation Tuesday.
"We're embarrassed, and he's embarrassed," manager Mike Matheny said, according to the St. Louis Post-Dispatch. "I'm not trying to shove my morality down anybody's throat. I've made that very clear. But I also need to be concerned about it [and] realize, is this something you'd want your kids to see? Is this something that you want to be known for? Is this how you want to be remembered?
"We are, like it or not, role models. Regardless of your moral compass and where it points, those sorts of things that we're talking about are not things we want our kids seeing."
Matheny said he has spoken with Martinez, who was "confused on what exactly happened" but "acknowledges, however, it did happen."
"He's willing to stand up and figure out how he's got to move forward," Matheny said, according to MLB.com.
On Wednesday, Martinez said his Twitter account was compromised.
"There was a confusion. My Twitter account was hacked," he told ESPNdeportes.com's Enrique Rojas. "It's my Twitter account and I understand what people are talking about and I should be responsible, but I want you to know that I do not have time to be doing that kind of thing.
"I talked to my teammates, the manager, the general manager. Everybody understood that I was a victim. My agent helped me, and everyone was aware that I did nothing wrong. One must be aware that there are people out there willing to hurt me."
For now, the Cardinals are using what happened as a learning experience.
On Tuesday, Matheny spoke with players in the team's major league camp for the third time this spring about social media, saying any miscue "is one fraction of a second away from being all over the world."
"It heightens our urgency to make sure that we're diligent and we're disciplined to educate and to make aware that when something like this happens, it's going to be an embarrassment to you, and when something like this happens, it's going to be an embarrassment to us," Matheny said, according to MLB.com.
What happened with Martinez offered the latest example.
"It's tricky," Cardinals GM John Mozeliak said when asked whether Martinez would be penalized, according to the Post-Dispatch. "This was not something that was negative to the club in the sense of [being] mean-spirited. It was just a poor choice of sites visited. It wasn't the smartest thing to do, but it happened and we hope that it doesn't happen again, and we hope other people learn from it."
Martinez says he has.
"I'm fighting a spot in the starting rotation, and that's what matters to me now," he told ESPNdeportes.com. "I don't have time to be on social networks doing bad things. These are problems that come to us and must be resolved quickly. From now on, I won't directly operate my account, I'll put a person in charge."
Information from ESPNdeportes.com's Enrique Rojas was used in this report.