It was the first time the running back answered questions about the Feb. 15 incident, which earned him a two-game suspension from the NFL.
"What happened that night is something that never should have happened," Rice, 27, said.
"It hurts because I can't go out there and play football, but it hurts more because I have to be a father and explain what happened to my daughter," he said.
Rice declined to say what triggered the incident, saying he just wants to move forward with his family.
"My actions that night were totally inexcusable," he said.
Rice called his wife an "angel" and said he let her, her parents, his teammates and the entire Baltimore community down.
He also brought up his mother.
"I know that's not who I am as a man," Rice said. "That's not who my mom raised me to be. If anyone knows me they know I was raised by a single parent and that was my mother."
He also said that "when the time is right," he and Palmer want to help other couples affected by domestic abuse. Rice called the violent fight a "one-time incident."
Rice was arrested following the altercation, in which he allegedly struck Palmer. Rice has been accepted into a diversion program, which upon completion could lead to the charges being expunged.
Last month, Rice met with NFL Commissioner Roger Goodell after joining the diversion program. Goodell ultimately decided to suspend the running back for two games without pay and fine him an additional game check "for conduct detrimental to the NFL in violation of the league's Personal Conduct Policy."
The Associated Press contributed to this report