Newcomb's apology was delivered in a meeting attended by players, coaches, manager Brian Snitker and Atlanta staff.
"I think Newcomb is one of the best teammates I've ever had,'' Inciarte said. "He's such a good person and a great example and role model for this organization, I'm more than proud to talk about the kind of person he is.''
Swanson, the shortstop who played with Newcomb in the Braves' minor league system before their time together in Atlanta, said he doesn't expect the incident to cause problems in the clubhouse.
"I think at the end of the day everyone realizes that does not directly reflect who he is,'' Swanson said of Newcomb's tweets. "He prides himself on being a great guy and a great teammate and we've got nothing but love for him. I think everybody was a little bit surprised but at the same time he addressed the team and we were able to handle things behind closed doors.''
Newcomb, 25, said he spoke Monday with Billy Bean, MLB's vice president for social responsibility and inclusion, and intends to meet with Bean when the Braves are in New York this week.
Bean also is expected to schedule a meeting with Washington Nationals shortstop Trea Turner after Turner's anti-gay and racially insensitive tweets from 2011 and 2012 surfaced Sunday night. The 25-year-old Turner apologized in a statement released by the Nationals.
Bean's meetings with Newcomb and Turner will determine appropriate sensitivity training.
Newcomb and Turner's tweets surfaced less than two weeks after years-old offensive tweets by Milwaukee Brewers reliever Josh Hader were brought to light during the All-Star Game. Hader apologized immediately afterward and was required to go through sensitivity training.
"It's been a tough couple of weeks for baseball on Twitter,'' Doolittle said. "It sucks to see racist and homophobic language coming from inside our league -- a league I'm so proud to be a part of that I've worked really hard to make a more accepting and inclusive place for all our fans to enjoy.''
Added Doolittle in his series of tweets on the subject: "It's entirely possible that those old posts no longer reflect that person's views. But actions will speak louder than words.''
Newcomb's tweets, posted in 2011 and 2012, included a racial epithet and several gay slurs, and emerged after he came within one out of a no-hitter in Sunday's 4-1 win over the Los Angeles Dodgers.
Newcomb is 10-5 with a 3.23 ERA in his breakout season. His near no-hitter was the season highlight that was quickly overshadowed by controversy.
"I hated it for him that probably the best day of his professional life, he went to bed thinking about this situation and it's not easy because he's a kid who cares,'' Snitker said. "He cares a lot and he's a good person and a great teammate. It was probably a rough night for him.''
Newcomb issued his first apology in an impromptu news conference less than an hour after Sunday's game and after his normal postgame interview.
He apologized again in another news conference following Monday's team meeting.
"I just want to say I'm sorry for the language that surfaced on my Twitter account,'' he said. "It's not acceptable in any way. It's not a reflection of the person or who I am. I don't think that's any kind of language that should be used in any context. ... I know it was hurtful and -- that's not a testament to who I am. I don't think it reflects who I am at all.''
The Associated Press contributed to this report.