It follows the arrest of The Member for Carmarthen East and Dinefwr on suspicion of assault at his home in Ammanford on 20 May.
In a written statement, Mr Edwards, 44, said: “I am deeply sorry. It is by far the biggest regret of my life.”
He was released on bail and a Plaid Cymru spokesman confirmed the whip had been withdrawn from Mr Edwards pending the conclusion of a police investigation.
Mr Edwards, who has not been charged, said: “I complied fully with the police and acted with the best interests of my wife and children as my primary motive throughout.
“I have referred myself to an investigation by my party.
“My priority now is to work with my wife to ensure as stable a future as possible for our family.”
His wife Emma Edwards also gave a statement, saying she accepted his apology.
She added: “Throughout the decade we have been together, he has been a loving and caring husband and father. As far as I am concerned the matter is now closed.”
Alun Ffred Jones, party chair of Plaid Cymru, said: “We are aware of the personal statement made by Mr Edwards, who recognises his actions fall below what is expected.
“Mr Edwards issued the statement with the support of his family and we ask that their privacy is respected.”