Jones took to Twitter on Saturday soon after Sam was drafted and tweeted "OMG" and "horrible." He later deleted the tweets.
He met with Dolphins brass about his comments Sunday, before the team ruled on his punishment. He must undergo educational training before he can return to the Dolphins, according to the team.
Jones apologized to Sam in his first public comments about the situation.
"I want to apologize to Michael Sam for the inappropriate comments that I made last night on social media," Jones said. "I take full responsibility for them, and I regret that these tweets took away from his draft moment. I remember last year when I was drafted in the seventh round and all of the emotions and happiness I felt when I received the call that gave me an opportunity to play for an NFL team, and I wish him all the best in his NFL career.
"I sincerely apologize to Mr. [Stephen] Ross, my teammates, coaches, staff and fans for these tweets. I am committed to represent the values of the Miami Dolphins organization and appreciate the opportunity I have been given to do so going forward."
Jones was drafted by the Dolphins in the seventh round in 2013. He was one of Miami's top special-teams players and finished with 10 tackles last season.
"We were disappointed to read Don's tweets during the NFL draft," Dolphins coach Joe Philbin said in a statement. "They were inappropriate and unacceptable, and we regret the negative impact these comments had on such an important weekend for the NFL. We met with Don today about respect, discrimination and judgment. These comments are not consistent with the values and standards of our program.
"We will continue to emphasize and educate our players that these statements will not be tolerated."
Jones became the second Dolphins player in the past week to make controversial statements on social media. After Miami drafted Ja'Wuan James in the first round Thursday, Pro Bowl center Mike Pouncey tweeted, "I can't wait for our gifts he's getting us." The comment was in poor taste in light of the Ted Wells report issued in February, when Pouncey and former teammates Richie Incognito and John Jerry were cited for bullying and harassing fellow lineman Jonathan Martin and others.
Pouncey shut down his Twitter account after meeting with team officials this weekend.
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